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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 GmbHISBNs: 3-527-27728-5 (Softcover); 3-527-60011-6 (Electronic)

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

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Printed in the Federal Republic of Germany.



Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV

1 Correlation between Energy and Mechanical Quantitiesof Face-Centred Cubic Metals, Cold-Worked and Softenedto Different States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Lothar Kaps, Frank Haeßner

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Experiments and Measurement Methods . . . . . . . . . . . . . . . . . . . . . . 182.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Cyclic stress-strain behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 Dislocation structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.3 Yield surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.3.1 Yield surfaces on AlMg3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.3.2 Yield surfaces on copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.3.3 Yield surfaces on steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Sequence Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

V



3 Plasticity of Metals and Life Prediction in the Rangeof Low-Cycle Fatigue: Description of DeformationBehaviour and Creep-Fatigue Interaction . . . . . . . . . . . . . . . . . . . 37Kyong-Tschong Rie, Henrik Wittke, Jurgen Olfe

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Experimental details for room-temperature tests . . . . . . . . . . . . . . . . 383.2.2 Experimental details for high-temperature tests . . . . . . . . . . . . . . . . . 393.3 Tests at Room Temperature: Description

of the Deformation Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Macroscopic test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.2 Microstructural results and interpretation . . . . . . . . . . . . . . . . . . . . . 433.3.3 Phenomenological description of the deformation behaviour . . . . . . . 453.3.3.1 Description of cyclic hardening curve, cyclic stress-strain curve

and hysteresis-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.3.2 Description of various hysteresis-loops with few constants . . . . . . . . . 473.3.4 Physically based description of deformation behaviour . . . . . . . . . . . 473.3.4.1 Internal stress measurement and cyclic proportional limit . . . . . . . . . . 473.3.4.2 Description of cyclic plasticity with the models

of Steck and Hatanaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3.5 Application in the field of fatigue-fracture mechanics . . . . . . . . . . . . 513.4 Creep-Fatigue Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.1 A physically based model for predicting LCF-life

under creep-fatigue interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.1.1 The original model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.1.2 Modifications of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.1.3 Experimental verification of the physical assumptions . . . . . . . . . . . . 553.4.1.4 Life prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4.2 Computer simulation and experimental verification

of cavity formation and growth during creep-fatigue . . . . . . . . . . . . . 573.4.2.1 Stereometric metallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.2.2 Computer simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4.3 In-situ measurement of local strain at the crack tip

during creep-fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.3.1 Influence of the crack length and the strain amplitude

on the local strain distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.3.2 Comparison of the strain field in tension and compression . . . . . . . . . 623.4.3.3 Influence of the hold time in tension on the strain field . . . . . . . . . . . 633.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Contents

VI

4 Development and Application of Constitutive Modelsfor the Plasticity of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Elmar Steck, Frank Thielecke, Malte Lewerenz

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 Mechanisms on the Microscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Simulation of the Development of Dislocation Structures . . . . . . . . . . 714.4 Stochastic Constitutive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.5 Material-Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.5.1 Characteristics of the inverse problem . . . . . . . . . . . . . . . . . . . . . . . 774.5.2 Multiple-shooting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.5.3 Hybrid optimization of costfunction . . . . . . . . . . . . . . . . . . . . . . . . . 774.5.4 Statistical analysis of estimates and experimental design . . . . . . . . . . 794.5.5 Parallelization and coupling with Finite-Element analysis . . . . . . . . . . 794.5.6 Comparison of experiments and simulations . . . . . . . . . . . . . . . . . . . 814.5.7 Consideration of experimental scattering . . . . . . . . . . . . . . . . . . . . . . 824.6 Finite-Element Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.6.1 Implementation and numerical treatment of the model equations . . . . 834.6.1.1 Transformation of the tensor-valued equations . . . . . . . . . . . . . . . . . . 844.6.1.2 Numerical integration of the differential equations . . . . . . . . . . . . . . . 854.6.1.3 Approximation of the tangent modulus . . . . . . . . . . . . . . . . . . . . . . . 864.6.2 Deformation behaviour of a notched specimen . . . . . . . . . . . . . . . . . 864.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 On the Physical Parameters Governing the Flow Stressof Solid Solutions in a Wide Range of Temperatures . . . . . . . . . . . 90Christoph Schwink, Ansgar Nortmann

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2 Solid Solution Strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.2.1 The critical resolved shear stress, �o . . . . . . . . . . . . . . . . . . . . . . . . . 925.2.2 The hardening shear stress, �d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3 Dynamic Strain Ageing (DSA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.3.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.3.2 Complete maps of stability boundaries . . . . . . . . . . . . . . . . . . . . . . . 945.3.3 Analysis of the processes inducing DSA . . . . . . . . . . . . . . . . . . . . . . 975.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.4 Summary and Relevance for the Collaborative Research Centre . . . . . 102

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Contents

VII

6 Inhomogeneity and Instability of Plastic Flowin Cu-Based Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Hartmut Neuhauser

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.2 Some Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.3 Deformation Processes around Room Temperature . . . . . . . . . . . . . . 1066.3.1 Development of single slip bands . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.3.2 Development of slip band bundles and Luders band propagation . . . . 1126.3.3 Comparison of single crystals and polycrystals . . . . . . . . . . . . . . . . . 1166.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.4 Deformation Processes at Intermediate Temperatures . . . . . . . . . . . . . 1186.4.1 Analysis of single stress serrations . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.4.2 Analysis of stress-time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.5 Deformation Processes at Elevated Temperatures . . . . . . . . . . . . . . . . 1246.5.1 Dynamical testing and stress relaxation . . . . . . . . . . . . . . . . . . . . . . . 1246.5.2 Creep experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128


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

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.2 The Model of Microscopic Structures . . . . . . . . . . . . . . . . . . . . . . . . 1317.2.1 The scale of observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.2.2 Basic slip mechanism in single crystals . . . . . . . . . . . . . . . . . . . . . . 1327.2.3 Treatment of polycrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.2.4 The Taylor theory in an appropriate version . . . . . . . . . . . . . . . . . . . 1337.3 Initial Orientation Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.3.1 Criteria of isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.3.2 Strategies for isotropic distributions . . . . . . . . . . . . . . . . . . . . . . . . . 1367.4 Numerical Calculation of Yield Surfaces . . . . . . . . . . . . . . . . . . . . . . 1377.5 Experimental Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.5.1 Prestraining of the specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.5.2 Yield-surface measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.5.3 Tensile test of a prestrained specimen . . . . . . . . . . . . . . . . . . . . . . . . 1427.5.4 Measured yield surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.5.5 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Contents

VIII

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

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.2 General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.3 The Deformation Law of Inelastic Solids . . . . . . . . . . . . . . . . . . . . . 1508.4 Bending of Rectangular Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.4.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.4.2 Experimental technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.4.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.4.3.1 Determination of the yield curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.4.3.2 Determination of the initial yield-locus curve . . . . . . . . . . . . . . . . . . 1588.5 Bending of Notched Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.5.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.5.2 Experimental technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1618.5.3 Approximation of displacement fields . . . . . . . . . . . . . . . . . . . . . . . . 1638.6 Identification of Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . 1658.6.1 Integration of the deformation law . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.6.2 Objective function, sensitivity analysis and optimization . . . . . . . . . . 1678.6.3 Results of parameter identification . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170


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

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1749.2 On Unified Models for Metallic Materials . . . . . . . . . . . . . . . . . . . . 1749.2.1 The overstress model by Chaboche and Rousselier . . . . . . . . . . . . . . 1759.2.2 Other unified models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1779.3 Time-Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1789.4 Adaptation of Model Parameters to Experimental Results . . . . . . . . . 1819.5 Systematic Approach to Improve Material Models . . . . . . . . . . . . . . . 1869.6 Models Employing Distorted Yield Surfaces . . . . . . . . . . . . . . . . . . . 1909.7 Approach to Cover Stochastic Test Results . . . . . . . . . . . . . . . . . . . . 1979.8 Structural Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2019.8.1 Consistent formulation of the coupled boundary

and initial value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2029.8.2 Analysis of stress-strain fields in welded joints . . . . . . . . . . . . . . . . . 2039.8.3 Thick-walled rotational vessel under inner pressure . . . . . . . . . . . . . . 205

Contents

IX

9.8.4 Application of distorted yield functions . . . . . . . . . . . . . . . . . . . . . . 2069.8.5 Application of the statistical approach of Section 9.7 . . . . . . . . . . . . . 2099.8.6 Numerical analysis for a recipient of a profile extrusion press . . . . . . 212


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

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21810.2 Material Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21910.2.1 Material, experimental set-ups, and techniques . . . . . . . . . . . . . . . . . 21910.2.2 Material behaviour under uniaxial cyclic loading . . . . . . . . . . . . . . . . 21910.2.2.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21910.2.2.2 Results of the uniaxial experiments . . . . . . . . . . . . . . . . . . . . . . . . . 22010.2.3 Material behaviour under biaxial cyclic loading . . . . . . . . . . . . . . . . 22510.2.3.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22510.2.3.2 Relations of tensile and torsional stresses . . . . . . . . . . . . . . . . . . . . . 22610.2.3.3 Yield-surface investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22910.3 Modelling of the Material Behaviour of Mild Steel Fe 510 . . . . . . . . 23610.3.1 Extended-two-surface model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23610.3.1.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23610.3.1.2 Loading and bounding surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23710.3.1.3 Strain-memory surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23810.3.1.4 Internal variables for the description on non-proportional loading . . . . 24110.3.1.5 Size of the yield surface under uniaxial cyclic plastic loading . . . . . . 24210.3.1.6 Size of the bounding surface under uniaxial cyclic plastic loading . . . 24210.3.1.7 Overshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24210.3.1.8 Additional update of �in in the case of biaxial loading . . . . . . . . . . . . 24310.3.1.9 Memory surface F� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24310.3.1.10 Additional isotropic deformation on the loading surface

due to non-proportional loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24410.3.1.11 Additional isotropic deformation of the bounding surface

due to non-proportional loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24410.3.2 Comparison between theory and experiments . . . . . . . . . . . . . . . . . . 24810.4 Experiments on Structural Components . . . . . . . . . . . . . . . . . . . . . . 24810.4.1 Experimental set-ups and computational method . . . . . . . . . . . . . . . . 24810.4.2 Correlation between experimental and theoretical results . . . . . . . . . . 24810.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

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11 Theoretical and Computational Shakedown Analysisof Non-Linear Kinematic Hardening Materialand Transition to Ductile Fracture . . . . . . . . . . . . . . . . . . . . . . . . 253Erwin Stein, Genbao Zhang, Yuejun Huang,Rolf Mahnken, Karin Wiechmann

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25311.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25311.1.1 General research topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25311.1.2 State of the art at the beginning of project B6 . . . . . . . . . . . . . . . . . . 25411.1.3 Aims and scope of project B6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25411.2 Review of the 3-D Overlay Model . . . . . . . . . . . . . . . . . . . . . . . . . . 25611.3 Numerical Approach to Shakedown Problems . . . . . . . . . . . . . . . . . . 25911.3.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25911.3.2 Perfectly plastic material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26011.3.2.1 The special SQP-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26011.3.2.2 A reduced basis technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26111.3.3 Unlimited kinematic hardening material . . . . . . . . . . . . . . . . . . . . . . 26111.3.4 Limited kinematic hardening material . . . . . . . . . . . . . . . . . . . . . . . . 26311.3.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26411.3.5.1 Thin-walled cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26411.3.5.2 Steel girder with a cope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.3.5.3 Incremental computations of shakedown limits

of cyclic kinematic hardening material . . . . . . . . . . . . . . . . . . . . . . . 26711.4 Transition to Ductile Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26911.5 Summary of the Main Results of Project B6 . . . . . . . . . . . . . . . . . . . 272

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

12 Parameter Identification for Inelastic Constitutive Equations Basedon Uniform and Non-Uniform Stress and Strain Distributions . . . 275Rolf Mahnken, Erwin Stein

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27512.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27512.1.1 State of the art at the beginning of project B8 . . . . . . . . . . . . . . . . . . 27512.1.2 Aims and scope of project B8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27612.2 Basic Terminology for Identification Problems . . . . . . . . . . . . . . . . . 27712.2.1 The direct problem: the state equation . . . . . . . . . . . . . . . . . . . . . . . 27712.2.2 The inverse problem: the least-squares problem . . . . . . . . . . . . . . . . . 27812.3 Parameter Identification for the Uniform Case . . . . . . . . . . . . . . . . . . 28012.3.1 Mathematical modelling of uniaxial visco-plastic problems . . . . . . . . 28012.3.2 Numerical solution of the direct problem . . . . . . . . . . . . . . . . . . . . . 28212.3.3 Numerical solution of the inverse problem . . . . . . . . . . . . . . . . . . . . 28212.4 Parameter Identification for the Non-Uniform Case . . . . . . . . . . . . . . 28312.4.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

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12.4.2 The direct problem: Galerkin weak form . . . . . . . . . . . . . . . . . . . . . 28512.4.3 The inverse problem: constrained least-squares optimization problem . 28612.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28712.5.1 Cyclic loading for AlMg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28712.5.2 Axisymmetric necking problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29012.6 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 294

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

13 Experimental Determination of Deformation- and Strain Fieldsby Optical Measuring Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 298Reinhold Ritter, Harald Friebe

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29813.2 Requirements of the Measuring Methods . . . . . . . . . . . . . . . . . . . . . 29813.3 Characteristics of the Optical Field-Measuring Methods . . . . . . . . . . . 29913.4 Object-Grating Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30013.4.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30013.4.2 Marking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30113.4.3 Deformation analysis at high temperatures . . . . . . . . . . . . . . . . . . . . 30213.4.4 Compensation of virtual deformation . . . . . . . . . . . . . . . . . . . . . . . . 30313.4.5 3-D deformation measuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30513.4.6 Specifications of the object-grating method . . . . . . . . . . . . . . . . . . . . 30513.5 Speckle Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30513.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30513.5.2 Technology of the Speckle interferometry . . . . . . . . . . . . . . . . . . . . . 30713.5.3 Specifications of the developed 3-D Speckle interferometer . . . . . . . . 30813.6 Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30913.6.1 2-D object-grating method in the high-temperature area . . . . . . . . . . . 30913.6.2 3-D object-grating method in fracture mechanics . . . . . . . . . . . . . . . . 30913.6.3 Speckle interferometry in welding . . . . . . . . . . . . . . . . . . . . . . . . . . 31013.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

14 Surface-Deformation Fields from Grating PicturesUsing Image Processing and Photogrammetry . . . . . . . . . . . . . . . . 318Klaus Andresen

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31814.2 Grating Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31914.2.1 Cross-correlation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31914.2.2 Line-following filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32114.3 3-D Coordinates by Imaging Functions . . . . . . . . . . . . . . . . . . . . . . . 32414.4 3-D Coordinates by Close-Range Photogrammetry . . . . . . . . . . . . . . 32514.4.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

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14.4.2 Parameters of the camera orientation . . . . . . . . . . . . . . . . . . . . . . . . 32614.4.3 3-D object coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32714.5 Displacement and Strain from an Object Grating: Plane Deformation . 32814.6 Strain for Large Spatial Deformation . . . . . . . . . . . . . . . . . . . . . . . . 32914.6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32914.6.2 Correcting the influence of curvature . . . . . . . . . . . . . . . . . . . . . . . . 33214.6.3 Simulation and numerical errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33314.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

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

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33715.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33915.2.1 Finite-Element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33915.2.1.1 Ambient temperature – rate-independent problem . . . . . . . . . . . . . . . 34015.2.1.2 Elevated temperature – visco-plastic problem . . . . . . . . . . . . . . . . . . 34115.2.2 Material models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34115.2.2.1 Ambient temperature – rate-independent problem . . . . . . . . . . . . . . . 34115.2.2.2 Elevated temperature – visco-plastic problem . . . . . . . . . . . . . . . . . . 34415.3 Experimental and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 34915.3.1 Test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34915.3.2 Computational analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34915.3.2.1 Monotonic loading – ambient temperature . . . . . . . . . . . . . . . . . . . . 35015.3.2.2 Cyclic loading – ambient temperature . . . . . . . . . . . . . . . . . . . . . . . . 35115.3.2.3 Time-dependent behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35615.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

16 Consideration of Inhomogeneities in the Applicationof Deformation Models, Describing the Inelastic Behaviourof Welded Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361Helmut Wohlfahrt, Dirk Brinkmann

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36116.2 Materials and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 36216.2.1 Materials and welded joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36216.2.2 Deformation models and numerical methods . . . . . . . . . . . . . . . . . . . 36516.2.2.1 Deformation model of Gerdes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36516.2.2.2 Fitting calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36516.3 Investigations with Homogeneous Structures . . . . . . . . . . . . . . . . . . . 365

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16.3.1 Experimental and numerical investigations . . . . . . . . . . . . . . . . . . . . 36616.3.1.1 Tensile tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36616.3.1.2 Creep tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36916.3.1.3 Cyclic tension-compression tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 37016.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37216.4 Investigations with Welded Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . 37416.4.1 Deformation behaviour of welded joints . . . . . . . . . . . . . . . . . . . . . . 37516.4.1.1 Experimental investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37516.4.1.2 Numerical investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37516.4.1.3 Finite-Element models of welded joints . . . . . . . . . . . . . . . . . . . . . . 37516.4.1.4 Calculation of the deformation behaviour of welded joints . . . . . . . . . 37516.4.2 Strain distributions of welded joints with broad weld seams . . . . . . . . 37616.4.3 Strain distributions of welded joints with small weld seams . . . . . . . . 38016.4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38016.5 Application Possibilities and Further Investigations . . . . . . . . . . . . . . 382

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

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

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Preface

The Collaborative Research Centre (Sonderforschungsbereich, SFB 319), “MaterialModels for the Inelastic Behaviour of Metallic Materials – Development and TechnicalApplication”, was supported by the Deutsche Forschungsgemeinschaft (DFG) from July1985 until the end of the year 1996. During this period of nearly 12 years, scientistsfrom the disciplines of metal physics, materials sciences, mechanics and applied engi-neering sciences cooperated with the aim to develop models for metallic materials on aphysically secured basis. The cooperation has resulted in a considerable improvementof the understanding between the different disciplines, in many new theoretical and ex-perimental methods and results, and in technically applicable constitutive models aswell as new knowledge concerning their application to practical engineering problems.

The cooperation within the SFB was supported by many contacts to scientists andengineers at other universities and research institutes in Germany as well as abroad.The authors of this report about the results of the SFB 319 wish to express their thanksto the Deutsche Forschungsgemeinschaft for the financial support and the very con-structive cooperation, and to all the colleagues who have contributed by their interestand their function as reviewers and advisors to the results of our research work.

Introduction

The development of mathematical models for the behaviour of technical materials is ofcourse directed towards their application in the practical engineering work. Besides theprojects, which have the technical application as their main goal, in all projects, whichwere involved in experiments with homogeneous or inhomogeneous test specimens –where partly also the numerical methods were further investigated and the implementa-tion of the material models in the programs was performed –, experiences concerningthe application of the models for practical problems could be gained. The whole-fieldmethods for measuring displacement and strain fields, which were developed in con-nection with these experiments, have given valuable support concerning the applicationof the developed constitutive models to practical engineering.

The research concerning the identification of the parameters of the models hasproven to be very actual. The investigations for most efficient methods for the param-eter identification will in the future still find considerable attention, where the coopera-tion of scientists from engineering as well as applied mathematics, which was started inthe SFB, will continue. As is shown in a later chapter, it is of increasing importance to

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use not only homogeneously, uniaxially loaded test specimen, but also to analyze stressand deformation fields in complexly loaded components. In connection with these in-vestigations, methods for the design of experiments should be developed, which can beused for the assessment of the structure of the material models and the physical mean-ing of the model parameters. The results obtained up to now have shown, also by com-parisons in cooperation with institutions outside the SFB, that the predictive propertiesof the developed material models are of equal quality as those of other models used inthe engineering practice. They have however the advantage that they are based on re-sults of material physics and therefore can use further developments of the knowledgeabout the mechanisms of inelastic deformations on the microscale.

During the work in the different projects, a surprising number of similar problemshave been found. Due to the close contacts between the working groups, they could beinvestigated with much higher quality than without this cooperation.

The exchange of thought between metal physics, materials sciences, mechanicsand applied engineering sciences was very stimulating and has resulted in the fact thatthe groups oriented towards application could be supported by the projects workingtheoretically, and on the other hand, the scientists working in theoretical fields couldobserve the application of their results in practical engineering.

Research Program

The main results of the activities of the SFB have been models for the load-deforma-tion behaviour as well as for damage development and the development of deformationanisotropies. These models make it possible to use results from the investigations frommetal physics and materials sciences in the SFB in the continuum mechanics models.The research work in metal physics and materials sciences has considerably contributedto a qualitative understanding of the processes, which have to be described by constitu-tive models. The structure of the developed models and of the formulations found inliterature, which have been considered for comparisons and supplementation of ourown development, have strongly influenced the work concerning the implementation ofthe material models in numerical computing methods and the treatment of technicalproblems. The models could be developed to a status, where the results of experimentalinvestigations can be used to determine the model parameters quantitatively.

This has resulted in an increasing activity on the experimental side of the workand also in an increase of the cooperation within the SFB and with institutions outsideof Braunschweig (BAM Berlin, TU Hamburg-Harburg, TH Darmstadt, RWTH Aachen,KFA Julich, KFZ Karlsruhe, Ecole Polytechnique Lausanne). In the SFB, joint researchwas undertaken in the fields of high-temperature experiments for the investigation ofcreep, cyclic loading and non-homogeneous stress and displacement fields for technicalimportant metallic materials, and their comparison with theoretical predictions. The de-veloped whole-field methods for measuring deformations have shown to be an impor-

Preface

XVI

tant experimental method. The increasing necessity to obtain experimental results ofhigh quality for testing and extending the material models has resulted in the develop-ment of experimental equipment, which also allows to investigate the material behav-iour under multiaxial loadings in the high- and low-temperature range.

The determination of model parameters and process quantities from experimentshas put the question for reliable methods for the parameter identification in the fore-ground. The earlier used methods of least-squares and probabilistic methods, such asthe evolution strategy, have given satisfying results. In the SFB, however, the know-ledge has developed that methods for the parameter identification, which consider thestructure of the material models and the design of optimal experiments and discriminat-ing experiments, deserve special consideration.

If numerical values for the model parameters are given, the possibility exists toexamine these values concerning their physical meaning, and in cooperation with thescientists from metal physics and materials sciences to investigate the connection be-tween the knowledge about the processes on the microscale and the macroscopic con-stitutive equations.

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

A: Materials behaviour• Phenomena• Material models• Parameter identification

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

down)

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 experi-mental investigations concerning the basis for the development of material models anddamage development from metal physics and materials sciences. In the following, ashort description of the activities within the research projects is given. Methods and re-sults are in detail given in later chapters.

Research Program

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Correlation between energetic and mechanical quantities of face-centred cubicmetals, cold-worked and softened to different states (Kaps, Haeßner)

One of these basic investigations is concerned with calorimetric measurements in con-nection with the description of recovery. After measurements based on the sheet rollingprocess, final investigations were performed concerning higher deformation tempera-tures and more complex deformation processes. Here, torsion experiments were exam-ined due to the fact that this process allows the investigation of very high deformationsas well as a simple reversal of the deformation direction and cyclic experiments.

Recovery and recrystallization are in direct competition with strain hardening. If amaterial is cold-worked, its yield stress increases. This process, denoted strain harden-ing, leads to a gain in internal energy. Recovery and recrystallization act to opposestrain hardening. Already upon deformation or during subsequent annealing, theseforces transform the material back into a state of lower energy. Although this reciproci-ty has been known for some time, the exact dependence of the process upon the typeand extent of deformation, upon the temperatures during deformation and softening an-neal as well as upon the chemical composition of the material is as yet only qualita-tively known. Consequently, the predictability of the processes is as poor as it has al-ways been so that, even today, one is still obliged to refer to experience and explicitexperiments for help.

Material state after uni- and biaxial cyclic deformation (Gieseke, Hillert, Lange)

The investigations concerning the material behaviour at multiaxial plastic deformationwere performed using the material AlMg3, copper and the austenitic stainless steel AISI316L. To find the connection between damage development and microstructure, the dis-locations structure at the tip of small cracks and at surface grains with differently pro-nounced slip-band development was investigated. With the aim to check the main as-sumptions of the two-surface models explicitly, measurements of the development ofthe yield surface of the material from the initial to the saturation state and within a sat-uration cycle were considerably extended. Consecutive yield surfaces along differentloading histories were measured. The two-surface models of Ellyin and McDowellwere implemented in the computations.

Technical components and structures today are increasingly being designed anddisplayed by computer-aided methods. High speed computers permit the use of mathe-matical models able to numerically reconstruct material behaviour, even in the courseof complex loading procedures.

In phenomenological continuum mechanics, the cyclic hardening and softeningbehaviour as well as the Bauschinger effect are described by yield-surface models. If aphysical formulation is chosen as a basis for these models, then it is vitally importantto have exact knowledge of the processes occurring in the metal lattice during deforma-tion. Two-surface models, going back to a development by Dafalias and Popov, de-scribe the displacement of the elastic deformation zone in a dual axis stress area. Theyield surfaces are assumed to be v. Mises shaped ellipses. However, from experimentswith uniaxial loading, it is known that the yield surfaces of small offset strains under

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load become characteristically deformed. In the present subproject, the effect of cyclicdeformation on the shape and position of the yield surfaces is studied, and their rela-tion to the dislocation structure is determined. To this end, the yield surfaces of threematerials with different slip behaviour were measured after prior uni- or biaxial defor-mation. The influence of the dislocation structures produced and the effect of internalstresses are discussed.

Plasticity of metals and life prediction in the range of low-cycle fatigue: description ofdeformation behaviour and creep-fatigue interaction (Rie, Wittke, Olfe)

In the field of investigations about the connection between creep and low-cycle fatigue,the development of models for predicting the componente lifetime at creep fatigue wasthe main aim of the work. Measuring the change of the physical magnitudes in themodel during an experiment results in an investigation and eventually a modification ofthe model assumptions. The model was also examined for its usability for experimentswith holding-times at the maximum pressure loading during a loading cycle.

For hot working tools, chemical plants, power plants, pressure vessels and tur-bines, one has to consider local plastic deformation at critical locations of structuralcomponents. Due to cyclic changes of temperature and load, the components are sub-jected to cyclic deformation, and the components are limited in their use by fatigue.After a quite small number of cycles with cyclic hardening or softening, a state of cyc-lic saturation is reached, which can be characterized by a stress-strain hysteresis-loop.Cyclic deformation in the regime of low-cycle fatigue (LCF) leads to the formation ofcracks, which can subsequently grow until failure of a component part takes place.

In the field of fatigue fracture mechanics, crack growth is correlated with param-eters, which take into account information especially about the steady-state stress-strainhysteresis-loops. Therefore, it can be expected that a more exact life prediction is possi-ble by a detailed investigation of the cyclic deformation behaviour and by the descrip-tion of the cyclic plasticity, e.g. with constitutive equations.

At high temperatures, creep deformation and creep damage are often superim-posed on the fatigue process. Therefore, in many cases, not one type of damage pre-vails, but the interaction of both fatigue and creep occurs, leading to failure of compo-nents.

The typical damage in the low-cycle fatigue regime is the development andgrowth of cracks. In the case of creep fatigue, grain boundary cavities may be formed,which interact with the propagating cracks, this leading to creep-fatigue interaction. Areliable life prediction model must consider this interaction.

The knowledge and description of the cavity formation and growth by means ofconstitutive equations are the basis for reliable life prediction. In the case of diffusion-con-trolled cavity growth, the distance between the voids has an important influence on theirgrowth. This occurs especially in the case of low-cycle fatigue, where the cavity formationplays an important role. Thus, the stochastic process of void nucleation on grain bound-aries and the cyclic dependence of this process has to be taken into consideration as atheoretical description. The experimental analysis has to detect the cavity-size distribu-tion, which is a consequence of the complex interaction between the cavities.

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Up to now, only macroscopic parameters such as the total stress and strain havebeen used for the calculation of the creep-fatigue damage. But crack growth is a localphenomenon, and the local conditions near the crack tip have to be taken into consid-eration. Therefore, the determination of the strain fields in front of cracks is an impor-tant 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 byslip processes in the crystal lattice, which are supported by the movement of lattice de-fects like dislocations and dislocation packages. The dislocation movements are op-posed by internal barriers, which have to be overcome by activation. This is performedby stresses or thermal energy. During the inelastic deformation, the dislocations interactand arrange in a hierarchy of structures such as walls, adders and cells. This forming ofinternal material structures influences strongly the macroscopic responses on mechani-cal and thermal loading.

A combination of models on the basis of molecular dynamics and cellular auto-mata is used to study numerically the forming of dislocation patterns and the evolutionof internal stresses during the deformation processes. For a realistic simulation, severalglide planes are considered, and for the calculation of the forces acting on a disloca-tion, a special extended neighbourhood is necessary. The study of the self-organizationprocesses with the developed simulation tool can result in valuable information for thechoice of formulations for the modelling of processes on the microscale.

The investigations concerning the development of material models based onmechanisms on the microscale have resulted in a unified stochastic model, which isable to represent essential and typical features of the low- and high-temperature plastic-ity. For the modelling of the dislocation movements in crystalline materials and theirtemperature and stress activation, a discrete Markov chain is considered. In order to de-scribe cyclic material behaviour, the widely accepted concept is used that the disloca-tion-gliding processes are driven by the effective stress as the difference between theapplied stress and the internal back stress. The influence of effective stress and tem-perature on the inelastic deformations is considered by a metalphysically motivatedevolution equation. A mean value formulation of this stochastic model leads to amacroscopic model consisting of non-linear ordinary differential equations. The resultsshow that the stochastic theory is helpful to deduce the properties of the macroscopicconstitutive equations from findings on the microscale.

Since the general form of the stochastic model must be adapted to the specialmaterial characteristics and the considered temperature regime, the identification of theunknown material parameters plays an important role for the application on numericalcalculations. The determination of the unknown material parameters is based on a Max-imum-Likelihood output-error method comparing experimental data to the numerical si-mulations. For the minimization of the costfunction, a hybrid optimization concept par-allelized with PVM is considered. It couples stochastic search procedures and severalNewton-type methods. A relative new approach for material parameter identification is

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the multiple shooting approach, which allows to make efficient use of additional measure-ment- and apriori-information about the states. This reduces the influence of bad initialparameters. Since replicated experiments for the same laboratory conditions show a sig-nificant scattering, these uncertainties must be taken into account for the parameter iden-tification. The reliability of the results can be tested with a statistical analysis.

Several different materials, like aluminium, copper, stainless steel AISI 304 andAISI 316, have been studied. For the analysis of structures, like a notched flat bar, theFinite-Element program ABAQUS is used in combination with the user material sub-routine UMAT. The simulations are compared with experimental data from gratingmethods.

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

In the area of the metal-physical foundations, investigations on poly- and single-crystal-line material have been performed. The superposition of solution hardening and ordi-nary hardening has found special consideration. Along the stress-strain curves, the lim-its between stable and unstable regions of deformation were investigated, and their de-pendencies on temperature, strain rate and solute concentration were determined. In re-gions of stable deformation, a quantitative analysis of the processes of dynamic strainageing (“Reckalterung”) was performed. The transition between regions of stable andunstable deformation was investigated and characterized.

At sufficiently low temperatures, host and solute atoms remain on their latticesites. The critical flow stress is governed by thermally activated dislocations glide (Ar-rhenius equation), which depends on an average activation enthalpy �G�0, and an effec-tive obstacle concentration cb. The total flow stress is composed of the critical flowstress and a hardening stress, which increases with the dislocation density in the cellwalls.

Detailed investigations on single crystals yielded expressions for the criticalresolved shear stress, �0 � �0��G�0� cb� T� ��, and the hardening shear stress,�d � �wGb�1�2

w . Here, �w is a constant, �w � 0�25 � 0�03, G the shear modulus, and�w the dislocation density inside the cell walls. The total shear stress results as� � �0 � �d.

At higher temperatures, the solutes become mobile in the lattice and cause an ad-ditional anchoring of the glide dislocations. This is described by an additional enthalpy�g�tw�Ea�� in the Arrhenius equation. In the main, it depends on the activation energyEa� of the diffusing solutes and the waiting time tw of the glide dislocations arrested atobstacles. Three different diffusion processes characterized by EaI�EaII�EaIII were foundfor the two f.c.c.-model systems investigated, CuMn and CuAl, respectively. In both,�g reaches values up to about 0.1 �G�0. Under certain conditions, the solute diffusioncauses instabilities in the flow stress, the well-known jerky flow phenomena (Portevin-Le Chatelier effect). Finally, above around 800 K in copper-based alloys, the solutes be-come freely mobile, and the critical flow stress as well as the additional enthalpy van-ish. In any temperature region, only a small total number of physical parameters is suf-ficient for modelling plastic deformation processes.

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Inhomogeneity and instability of plastic flow in Cu-based alloys (Neuhauser)

In a second project, the main goal of the research is to clarify the physical mecha-nisms, which control the kinetics of the deformation, especially in such parameter re-gions, which are characterized by inhomogeneity and instability of the deformation pro-cess. It is looked for a realistic interpretation of the magnitudes, which will be usedwith empirical material equations as it is necessary for a sensible application and extra-polation to extended parameter regions. Especially, reasons and effects of deformation-inhomogeneities and -instabilities in the systems Cu-Al and Cu-Mn, which show ten-dencies to short-range order, were investigated. Determining dislocation-generationrates and dislocation velocities in the case of gradients of the effective stress were aswell aim of the investigations as the influence of diffusion processes on the generation(blocking, break-away) and motion (obstacle destruction and regeneration) of disloca-tions. Investigations were also performed concerning the use of the results for singlecrystals for the description of the practically more important case of the behaviour ofpolycrystals. In this case, especially the influence of the grain-boundaries on generationand movement of dislocations or dislocation groups has to be considered.

The special technique used in this project is a microcinematographic method,which permits to measure the local strain and strain rate in slip bands, which are theactive regions of the crystal. Cu-based alloys with several percent of Al and Mn solutesare considered in order to separate the effects of stacking-fault energy from those of so-lute hardening and short-range ordering, which are comparable for both alloy systems,while the stacking-fault energy decreases rapidly with solute concentration for CuAlcontrary to CuMn alloys. Both systems show different degrees of inhomogeneous slipin the length scales from nm to mm (slip bands, Luders bands), and, in a certain rangeof deformation conditions, macroscopic deformation instabilities (Portevin-Le Chateliereffect). These effects have been studied in particular.

The influence of large torsional prestrain on the texture development and yield surfaceof polycrystals – experimental and theoretical investigations (Besdo, Wellerdick-Wojta-sik)

This research project consists of a theoretical and an experimental part. The topic ofthe theoretical part was the simulation of texture development and methods of calculat-ing yield surfaces. The calculations started from an initially isotropic grain distribution.Therefore, it was necessary to set up such a distribution. Different possibilities werecompared with an isotropy test considering the elastic and plastic properties. With somefinal distributions, numerical calculations were carried out. The Taylor theory in an ap-propriate version and a simple formulation based on the Sachs assumption were used.

Calculation of yield surfaces from texture data can be done in many different ways.Some examples are the yield surfaces calculated with the Taylor theory, averaging meth-ods or formulations, which take the elastic behaviour into account. Several possibilitiesare presented, and the numerical calculations are compared with the experimental results.

In order to measure yield surfaces after large torsional prestrain, thin-walled tubu-lar specimens of AlMg3 were loaded up to a shear strain of � � 1�5, while torsional

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buckling was prevented by inserting a greased mandrel inside the specimens. Furtherinvestigations of the prestrained specimens were done with the testing machine of theproject area B.

At least one yield surface, represented by 16 yield points, was measured witheach specimen. The yield point is defined by the offset-strain definition, where gener-ally the von Mises equivalent offset strain is used. Three different loading paths wererealized with the extension-controlled testing machine. Thus, the results were yield sur-faces measured with different offsets and loading paths.

The offset-strain definition is based on the elastic tensile and shear modulus.These constants were calculated at the beginning of each loading path, and since theystrongly effect the yield surfaces, this must be done with the highest amount of care.The isotropic specimens are insensitive to different loading paths, and the measuredyield surfaces seem to be of the von Mises type. By contrast, the prestrained specimensare very sensitive to different loading paths. Especially the shape and the distorsion ofthe measured surfaces changes as a result of the small plastic strain during the measure-ment. Therefore, it seems that the shape and the distortion of the yield surface were notstrongly effected by the texture of the material.

Parameter identification of inelastic deformation laws analysing inhomogeneous stress-strain states (Kreißig, Naumann, Benedix, Borman, Grewolls, Kretzschmar)

In the last years, the necessity of solutions of non-linear solid mechanics problems haspermanently increased. Although powerful hard- and software exist for such problems,often more or less large differences between numerical and experimental results are ob-served. The dominant reason for these defects must be seen in the material-dependentpart of the used computer programs. Either suitable deformation laws are not imple-mented or the required parameters are missing.

Experiments on the material behaviour are commonly realized for homogeneousstress-strain states, as for example the uniaxial tensile and compression test or the thin-walled tube under combined torsion, tensile and internal pressure loading. In additionto these well-known methods, experimental studies of inhomogeneous strain and stressfields are an interesting alternative to identify material parameters.

Two types of specimens have been investigated. Unnotched bending specimenshave been used to determine the elastic constants, the initial yield locus curve and theuniaxial tension and compression yield curves. Notched bending specimens allow ex-periments on the hardening behaviour due to inhomogeneous stress-strain states.

The numerical analysis has been carried out by the integration of the deformationlaw at a certain number of comparative points of the ligament with strain increments,determined from Moire fringe patterns, as loads. The identification of material param-eters has been performed by the minimization of a least-squares functional using deter-ministic gradient-type methods. As comparative quantities have been taken into accountthe bending moment, the normal force and the stresses at the notch grooves.

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Project area B: development of computational methods

The essential goals of the project area B were the transfer of experimental results inmaterial models, which describe the essential characteristics of the complex non-linearbehaviour of metallic materials in a technically satisfactory manner. For this reason,known formulations of material models, developments of the SFB and new formula-tions 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 modelscontain internal variables, which can either be purely phenomenological or be based onmicrostructural considerations. In the frame of the SFB, the goal was the microstructur-al substantiation of these internal variables.

For the adjustment of the model parameters on the experimental results, optimiza-tion strategies are necessary, which allow judging the power of the models. The ob-tained results showed that this question is of high importance, also for further research.Extensions for multiaxial loading cases have been developed and validated. For the in-vestigated loadings of metals at high temperatures and alternating and cyclic loadinghistories as well as for significantly time-dependent material behaviour, the literatureshows only a first beginning in the research concerning such extensions.

The material models had firstly to be examined concerning the materials. For thepractical application, however, their suitability for their implementation in numerical al-gorithms (e.g. Finite-Element methods) and the influence on the efficiency of numeri-cal computations had to be examined.

Especially for the computation of time-dependent processes, numerically stableand – because of the expensive numerical calculations – efficient computational algo-rithms had to be developed (e.g. fast converging time-integration methods for stronglynon-linear problems).

The developed (or chosen) material models and algorithms had to be applied forlarger structures, not only to test the computational models, but simultaneously also –by reflection to the assumptions in the material models – to find out which parametersare of essential meaning for the practical application, and which are rather unimportantand can be neglected. This results in the necessity to perform on all levels sensitivityinvestigations for the relevancy of the variants of the assumptions and their parameters.

At loading histories, which describe alternating or cyclic processes due to the al-ternating plastification, the question of saturation of the stress-strain histories andshakedown are of special importance. The projects in the project area B were investi-gating these problems in a complementary manner. They were important, central ques-tions conceived so that related problems were investigated to accelerate the progress ofthe work and to allow mutual support and critical exchange of thought.

Development and improvement of unified models and applications to structural analysis(Ahrens, Duddeck, Kowalsky, Pensky, Streilein)

Especially for structures of large damage potentials, the design has to simulate failureconditions as realistic as possible. Therefore, inelastic and time-dependent behavioursuch as temperature-induced creep have to be considered. Besides adequate numerical

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methods of analyses (as non-linear Finite-Element methods), mathematically correctmodels are needed for the thermal-mechanical material behaviour under complex load-ings. Unified models for metallic materials cover time-independent as well as time-de-pendent reactions by a unified concept of elasto-viscoplasticity.

Research results are presented, which demonstrate further developments for uni-fied models in three different aspects. The methodical approach is shown firstly on thelevel of the material model. Then, verifications of their applicability are given by utiliz-ing them in the analyses of structures. The three aspects are the following problems:

1. Discrepancies between results of experimental and numerical material behaviourmay be caused by• insufficient or inaccurate parameters of the material model,• inadequate material functions of the unified models,• insufficient basic formulations for the physical properties covered by the model.It is shown that more consistent formulations can be achieved for all these threesources of deficits by systematic numerical investigations.

2. Most of the models for metallic materials assume yield functions of the v. Misestype. For hardening, isotropic and/or kinematic evolutions are developed, that corre-spond to affine expansions or simple shifting of the original yield surface, whereasexperimental results show a distinctive change of the shape of the yield surfaces(rotated or dented) depending on the load path. To cover this material behaviour ofdistorted yield surfaces, a hierarchical expansion of the hardening rule is proposed.The evolutionary equations of the hardening (expressed in tensors) are extended byincluding higher order terms of the tensorial expressions.

3. Even very accurately repeated tests of the same charge of a metallic material show acertain scattering distribution of the experimental results. The investigation of testseries (provided by other projects of the SFB) proved that a normal Gaussian distri-bution can be assumed. A systematic approach is proposed to deal with such experi-mental deviations in evaluating the parameters of the material model.

The concepts in all of the three items are valid in general although the overstressmodel by Chaboche and Rousselier is chosen here for convenience.

In verifying the conceptual improvements, it is necessary to provide accurate andefficient procedures for time-integration processes and for the evaluation of the modelparameters via optimization. In both cases, different procedures are elaborately com-pared with each other.

Results of the numerical analyses of different structures are given. They demon-strate the efficiency of the proposed further developments by applying Finite-Elementmethods for non-linear stress-displacement problems. This includes:

• investigations of welded joints with modifications of the layers of different micro-structures,

• thick-walled vessels in order to demonstrate the effects of different formulations ofthe material model on the stress-deformation fields of larger structures,

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• distorted yield functions to a plate with an opening,• effects of stochastic distribution of material behaviour to a plate with openings,• the application of material models based on microphysical mechanisms to a larger

vessel, the recipient of hot aluminium blocks for a profile extrusion press.

On the behaviour of mild steel Fe 510 under complex cyclic loading (Peil, Scheer,Scheibe, Reininghaus, Kuck, Dannemeyer)

The employment of the plastic bearing capacity of structures has been recently allowedin both national and international steel constructions standards. The ductile material be-haviour of mild steel allows a load-increase well over the elastic limit. To make use ofthis effect, efficient algorithms, taking account of the plastic behaviour under cyclic orrandom loads in particular, are an important prerequisite for a precise calculation of thestructure.

The basic elements of a time-independent material model, which allows to takeinto account the biaxial or random load history for a mild steel under room tempera-ture, are presented. In a first step, the material response under cyclic or random loadshas to be determined. The fundamentals of an extended-two-surface model based onthe two-surface model of Dafalias and Popov are presented. The adaptations have beenmade in accordance with the results of experiments under multiaxial cyclic loadings.Finally, tests on structural components are performed to verify the results obtainedfrom the calculations with the described model.

Theoretical and computational shakedown analysis of non-linear kinematic hardeningmaterial and transition to ductile fracture (Stein, Zhang, Huang, Mahnken, Wiechmann)

The response of an elastic-plastic system subjected to variable loadings can be verycomplicated. If the applied loads are small enough, the system will remain elastic forall possible loads. Whereas if the ultimate load of the system is attained, a collapsemechanism will develop and the system will fail due to infinitely growing displace-ments. Besides this, there are three different steady states, that can be reached while theloading proceeds:

1. Incremental failure occurs if at some points or parts of the system, the remainingdisplacements and strains accumulate during a change of loading. The system willfail due to the fact that the initial geometry is lost.

2. Alternating plasticity occurs, this means that the sign of the increment of the plasticdeformation during one load cycle is changing alternately. Though the remainingdisplacements are bounded, plastification will not cease, and the system fails locally.

3. Elastic shakedown occurs if after initial yielding plastification subsides, and the sys-tem behaves elastically due to the fact that a stationary residual stress field isformed, and the total dissipated energy becomes stationary. Elastic shakedown (or

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

The research work is based on Melan’s static shakedown theorems for perfectly plasticand linear kinematic hardening materials, and is extended to generally non-linear limitedhardening by a so-called overlay model, being the 3-D generalization of Neal’s 1-D mod-el, for which a theorem and a corollary are derived. Finite-Element method and adequateoptimization algorithms are used for numerical approach of 2-D problems. A new lemmaallows for the distinction between local and global failure. Some numerical examples il-lustrate the theoretical results. The shakedown behaviour of a cracked ductile body is in-vestigated, where a crack is treated as a sharp notch. Thresholds for no crack propagationare formulated based on shakedown theory.

Parameter identification for inelastic constitutive equations based on uniform and non-uniform stress and strain distributions (Mahnken, Stein)

In this project, various aspects for identification of parameters are discussed. Firstly, asin classical strategies, a least-squares functional is minimized using data of specimenwith stresses and strains assumed to be uniform within the whole volume of the sam-ple. Furthermore, in order to account for possible non-uniformness of stress and straindistributions, identification is performed with the Finite-Element method, where alsothe geometrically non-linear case is taken into account. In both approaches, gradient-based optimization strategies are applied, where the associated sensitivity analysis isperformed in a systematic manner. Numerical examples for the uniform case are pre-sented with a material model due to Chaboche with cyclic loading. For the non-uni-form case, material parameters are obtained for a multiplicative plasticity model, whereexperimental data are determined with a grating method for an axisymmetric neckingproblem. In both examples, the results are discussed when different starting values areused and stochastic perturbations of the experimental data are applied.

Project area C: experimental verification

Material parameters, which describe the inelastic behaviour of metallic materials, canbe determined experimentally from the deformation of a test specimen by suitable cho-sen basic experiments. One-dimensional load-displacement measurements, however, arenot providing sufficient informations to identify parameters of three-dimensionalmaterial laws. For this purpose, the complete whole-field deformation respectivelystrain state of the considered object surface is needed. It can be measured by opticalmethods. They yield the displacement distribution in three dimensions and the straincomponents in two dimensions. So, these methods make possible an extensive compari-son of the results of a related Finite-Element computation.

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Experimental determination of deformation- and strain fields by optical measuringmethods (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 theconsidered object surface marked by an attached grating, consisting of a deterministicor stochastic grey value distribution, and recorded by the photogrammetric principle.Then, the strain follows from the difference of the displacement vectors of two neigh-bouring points related to two different deformation states of the object and related totheir initial distance.

The electronic Speckle interferometry is based on the Speckle effect. It comesinto existence if an optical rough object surface is illuminated by coherent light, andthe scattered waves interfere. By superposing of the interference effects of an objectand reference wave related to two different object states, the difference of the arisingSpeckle patterns leads to correlation fringes, which describe the displacement field ofthe considered object.

Regarding the object-grating method, grating structures and their attachment havebeen developed, which can be analysed automatically and which are practicable also athigh temperatures up to 1000 �C, as often inelastic processes take place under this con-dition. Furthermore, the optical set-up, based on the photogrammetric principle, wasadapted to the short-range field with testing fields of only a few square millimeters.The object-grating method is applicable if the strain values are greater than 0.1%.

For measurement of smaller strain values down to 10–5, the Speckle interfero-metric principle was applied. A 3-D electronic Speckle interferometer has been devel-oped, which is so small that it can be adapted directly at a testing machine. It is basedon the well-known path of rays of the Speckle interferometry including modern opto-electronic components as laser diodes, piezo crystals and CCD-cameras.

Furthermore, both methods are suitable for high resolution of a large change ofmaterial behaviour. Finally, the measurement can be conducted at the original and takesplace without contact and interaction.

Surface deformation fields from grating pictures using image processing and photo-grammetry (Andresen)

The before-mentioned grating techniques are optical whole-field methods applied to de-rive the shape or the displacement and strain on the surface of an object. A regulargrating fixed or projected on the surface is moved or deformed together with the ob-ject. In different states, pictures are taken by film cameras or by electronic cameras.For plane surfaces parallel to the image plane, one camera supplies the necessary infor-mation for displacement and strain. To get the spatial coordinates of curved surfaces,two or more stereocameras must be used. In early times, the grating patterns were eval-uated manually by projecting the images to large screens or by use of microscope tech-niques. Today, the pictures are usually digitized, yielding resolutions from 200 ×200 to2000 ×2000 picture elements (pixels or pels) with generally 256 grey levels (8 bit). By

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suitable image-processing methods, the grating coordinates in the images are deter-mined to a large extent automatically. The corresponding coordinates on plane objectsare derived from the image coordinates by a perspective transformation. Consideringspatial surfaces, first, the orientation of the cameras in space must be determined by acalibration procedure. Then, the spatial coordinates are given by intersection of the raysof adjoined grating points in the images.

The sequence of the grating coordinates in different states describes displacementand strain of the considered object surface. Applying suitable interpolation gives contin-uous fields for the geometrical and physical quantities on the surface. These experimen-tally determined fields are used for

• getting insight into two-dimensional deformation processes and effects,• supplying experimental data to the theoretically working scientist,• providing experimental data to be compared with Finite-Element methods,• deriving parameters in standard constitutive laws,• developing constitutive laws with new dependencies and parameters.

Experimental and numerical analysis of the inelastic postbuckling behaviour of shear-loaded aluminium panels (Kossira, Arnst)

As a practical problem of aircraft engineering, the case of shear-loaded thin panels outof the material AlCuMg2 under cyclic, quasistatic loading was investigated by experi-mental and numerical methods. Beyond the up-to-now used classical theory of plastici-ty, the theoretical research was based on the “unified” models, which were developedand adjusted to numerical computational methods in other areas of the research project.

Shear-loaded panels are in general substructures of aerospace constructions sincethere are always load cases during a flight mission, in which shear loads are predomi-nant in the thin-walled structures of subsonic as well as in supersonic and hypersonicaircrafts. The good-natured postcritical load-carrying behaviour of shear-loaded panelsat moderate plastic deformations can be exploited in emergency (fail safe) cases sincethey exhibit no dramatic loss of stiffness even in the high plastic postbuckling regime.The temperature at the surface of hypersonic vehicles may reach very high values, butwith a thermal protection shield, the temperatures of the load-carrying structure can bereduced to moderate values, which allow the application of aluminium alloys. There-fore, the properties of the mostly used aluminium alloy 2024-T3 are taken as a basisfor the experimental and theoretical studies of the behaviour of shear-loaded panels atroom temperature and at 200 �C.

The primary aim of these studies is the understanding of the occurring phenom-ena, respectively the examination of the load-carrying behaviour of the consideredstructures under different load-time histories, and to provide suitable data for the de-sign. Besides experimental investigations, which are achieved by a specially designedtest set-up, the development of numerical methods, which describe the phenomena, wasnecessary to accomplish this intention. The used numerical model is based on a Finite-Element method, which is capable of calculating the geometric and physical non-linear– in case of visco-plastic material behaviour time-dependent – postbuckling behaviour.

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A substantial problem within the numerical method was the simulation of the non-lin-ear material properties. Using a rate-independent two-surface material model and amodified visco-plastic material model of the Chaboche type, the non-linear propertiesof the aluminium alloy 2024-T3 are approximated with sufficient accuracy at both con-sidered temperatures.

Some results of the theoretical and experimental studies on the monotonic andcyclic postbuckling behaviour of thin-walled aluminium panels under shear load at am-bient and elevated temperatures are presented. The applied loads exceed the theoreticalbuckling loads by factors up to 40, accompanied by the occurrence of moderate inelas-tic deformations. Apart from the numerical model, the monotonic loading, subsequentcreep rates, the snap-through behaviour at cyclic loading, the inelastic processes duringloading, and the influence of the aspect ratio are major topics in the presented discus-sion of the results for shear-loaded panels at room temperature and at 200 �C.

Consideration of inhomogeneities in the application of deformation models, describingthe inelastic behaviour of welded joints (Wohlfahrt, Brinkmann)

A second practical problem was the investigation of the influence of welded joints onthe mechanical behaviour of components, which is due to the high degree of “Werk-stoffnutzung” in modern welded structures of high importance. Special considerationwas given here to the important question of the material behaviour at cyclic loading aswell from the point of view of numerical computation of these processes and the con-nected effects as from the point of view of the problems connected with aspects ofmaterials sciences.

The local loads and deformations in welded joints have rarely been investigatedunder the aspect that the mechanical behaviour is influenced by different kinds of mi-crostructure. These different kinds of microstructure lead to multiaxial states of stressesand strains, and some investigations have shown that for the determination of the totalstate of deformation of a welded joint, the locally different deformation behaviour hasto be taken into account. It is also published that different mechanical properties in theheat-affected zone as well as a weld metal with a lower strength than the base metalcan be the reason or the starting point of a fracture in welded joints. A new investiga-tion demonstrates that in TIG-welded joints of the high strength steel StE690, a fine-grained area in the heat-affected zone with a lower strength than that of the base metalis exclusively the starting zone of fracture under cyclic loading in the fully compressiverange. These investigations support the approach described here that the mechanical be-haviour of the different kinds of microstructure in the heat-affected zone of weldedjoints has to be taken into account in the deformation analysis. The influences of theseinhomogeneities on the local deformation behaviour of welded joints were determinedby experiments and numerical calculations over a wide range of temperature and load-ing. The numerical deformation analysis was performed with ttformat

he method of Finite-Elements, in which recently developed deformation modelssimulate the mechanical behaviour of materials over the tested range of temperatureand loading conditions.

Preface

XXX

1 Correlation between Energeticand Mechanical Quantities of Face-Centred CubicMetals, 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 dependon the material as well as on the type and degree of deformation. The kinetic parame-ters can in principle be determined by calorimetric methods. By combining calorimetri-cally determined values with characteristics measured mechanically and with micro-structural data, information can be gained about the strain-hardened state and the mech-anism of the softening process.

This materials information can support critical assessment of the structure ofmaterial models and hence be utilized for the appropriate adjustment of constitutivemodels to material properties.

1.2 Experiments

One objective of the work in this particular area of research was to investigate the de-pendence of the softening kinetics of face-centred cubic metals on the deformation. Thechosen types of deformation were torsion, tension and rolling. In the cases of torsionand tension, additional cyclic experiments with plastic amplitudes of 0.01 to 0.1 werecarried out. The materials studied were aluminium, lead, nickel, copper and silver.Thus, in this order, metals of very high to very low stacking fault energy were investi-gated. In the following presentation of the results, the emphasis will be on copper.

To determine the mechanical data, the first step was to characterize the deforma-tion with the aid of the crystallographic slip �, the shear stress �N normalized to the

1

* Technische Universitat Braunschweig, Institut fur Werkstoffe, Langer Kamp 8,D-38106 Braunschweig, Germany



shearing modulus G � �N � �c�G, and the strain-hardening rate � � ��N��a. The con-version 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 subjectedto torsion at various temperatures. The characterization is clear because for increasingdeformation temperature, a decreasing yield stress results.

Figure 1.2 shows the strain-hardening rate versus the normalized shear stress ofcopper to extreme deformation. The strain-hardening rate can be subdivided into threeregions, which, following the literature, may be denoted strain-hardening regions III toV [3]. Regions III and V show a linearly decreasing strain-hardening rate with shearstress. Region IV, as region II, is characterized by constant strain hardening.

The occurrence of these different regions depends strongly on the type of deforma-tion. Thus, for tensile deformation, in consequence of instability, only deformation to re-gion III can be realized. Rolling permits greater deformation, but brings with it the problemof defining a specific measurement to categorize the strain-hardening regions. The tempera-ture effect of the deformation fits well into the scheme proposed by Gil Sevillano [4].According to this scheme, all flow curves in region III may be described by a fixed initialstrain-hardening rate �III

0 and a variable limiting stress �IIIS . This latter is affected by dy-

namic recovery and is therefore dependent on deformation temperature and velocity. Itdecreases for increasing deformation temperature and increases for higher deformationvelocities.

This statement is also true for the other characteristic stresses �IV� �V� �VS � The loga-

rithm of the characteristic stress decreases linearly with the normalized deformationtemperature, TN � kT�Gb3� The normalization was proposed by Mecking et al. [5]. It

1 Correlation between Energetic and Mechanical Quantities

2

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

has been successfully applied to our own measurements. However, it may be seen thatthe dependence on temperature is different for the individual stresses (Figure 1.3).

Careful evaluation of the experiments taking account of the effects of texture andsample shows similarities as well as differences between the two deformation types ten-sion and torsion. Up to a slip value of a=0.4, the flow curve shows little difference be-tween tension and torsion. Above that value, the hardening is greater for the tension ex-periment (Figure 1.4).

The differences are more pronounced when the hardening rate is studied ratherthan the flow curve. From the start, the former lies higher for tension than for torsion.The different procedures may be followed microstructurally using a transmission elec-

1.2 Experiments

3

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

Figure 1.3: Characteristic stresses for the strain hardening of copper versus the normalized tem-perature.

tron microscope. Other authors have described this influence of the load path on themicrostructure [6–9]. The reason for this may be that different average numbers of slipsystems are necessary for deformation [10]. This also affects the development of activa-tion energies �G0 and activation volumes V. To determine these quantities, velocitiesare varied in tension and torsion experiments, i.e. during a unidirectional experiment,the extension rate is momentarily increased. In those sections with an increased exten-sion rate, the material shows a higher flow stress. For the evaluation, the following an-satz was chosen for the relationship between the extension rate �� and the flow stress �:

�� 0 exp��G0 � V��

kT

� �� 1�

The activation volume and energy are the important quantities for the constitutive equa-tions developed in the subproject A6 [11, 12]. The comparison of the deformationtypes tension and torsion shows a definite difference in the development of activationvolumes with �N. This is manifest by the tension (strain) deformation, which exhibits aconstant velocity sensitivity even for significantly smaller degrees of deformation (Fig-ure 1.5). The activation volumes are a particularly indicative measurement for thevelocity sensitivity. In region III for torsion, they show a continuous decrease, whichbecomes less only upon reaching region IV. For tension, on the other hand, there arealso two sections with decreasing or nearly constant activation volumes. However, thetransition in the curve of the activation volume versus the stress already lies in thestrain-hardening region III.

The activation energies �G0 for torsion were determined from the characteristicstresses for different temperatures (cf. Figure 1.3). The resultant values for the stresses�III

S � �IVand �V

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 de-

termined using a calorimeter. As expected, the stored energy shows a monotonic in-


4

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

crease with deformation. Moreover, dynamic recovery counteracts energy storage as itdoes hardening. Hence, there is an unequivocal correlation between the deformationtemperature and stored energy such that an increasing deformation temperature leads toless stored energy (Figure 1.6).

Figure 1.6 demonstrates the great influence of the stacking fault energy. The valueof the reduced stacking fault energy for silver lies at 2.4·10–3 compared with the valueof 4.7·10–3 for copper. Lower stacking fault energies lead to a greater separation of par-

1.2 Experiments

5

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

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

tial dislocations. This hinders dynamic recovery because the mechanism of cross slip isimpaired.

The connection between stored energy and shearing stress was studied for defor-mation by torsion, tension and push-pull. There is a clear tendency to store more en-ergy with increasing deformation temperature at constant shearing stress. It would ap-pear that energy storage by more fully condensed states is more effective. The mea-sured values for tension and push-pull in this sequence lie above those for the greatesttorsional deformation. For the same shearing stress, silver also clearly stores more en-ergy than copper.

This relationship is represented in the Figures 1.7 and 1.8. Figure 1.7 comprisestorsion experiments up to extreme deformation. Figure 1.8 shows a comparison of vari-ous types of deformation. For better resolution, the abscissa here is confined to smalland intermediate values of stress. The variable behaviour of the materials and the effectof the types of deformation may also be demonstrated in measurements of the soften-ing kinetics to be discussed. In analogy to the strain-hardening rate, an energy storagerate �E � �ES��N has been defined. This quantity represents independent information.

The development of the energy storage rate is clearly correlated with the strain-hardening stages (Figure 1.9). The combination of energetic and mechanical measure-ments permits a statement on the change in dislocation density �, to a first approxima-tion proportional to the stored energy, with increasing flow stress. A linearly increasingenergy storage rate with stress leads to a law of the type:

�ES

��N��N � �c

G� k1

��ES

�� 2�

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


6

Figure 1.7: Stored energy versus normalized shear stress for copper and silver deformed at differ-ent temperatures.

with a different proportionality factor � of value rather below the one pertaining to re-gion III. The factor � may only be analytically assessed for deformation in the regionof the strain-hardening stage II. For greater plastic deformation, which would then bedeformation in region III of the strain-hardening stage, this factor is of a qualitative na-ture. The evolution of � for various materials, deformation temperatures and types ofdeformation is collated in Table 1.1. The stress in the second column indicates the endof the linear storage rate in the strain-hardening region III.

1.2 Experiments

7

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

Figure 1.9: Stored energy (upper curve) and rate of energy storage of distorted copper versus nor-malized shear stress.

� Cu 235 K s Cu 293 K � Cu 373 K + Cu 293 K Tension × Cu 293 K Push-Pull

If X denotes the softened fraction of the material, one may attempt to describe thesoftening kinetics �X by a product of functions, which combines the thermal activationand the nature of the reaction in one appropriate multiplier:

�X � f �X�g�T� � f �X� exp � QRT

� �� 3�

Equation (3) is easily handled numerically. The activation energy of the softening Qand the form function f may be determined separately. Equation (3) offers the added ad-vantage that, as a rate equation, it may be directly incorporated into a constitutive equa-tion if the quantities Q and f �X� are known. The simpler analysis considers the productand in its place the reaction temperature. This temperature is a direct measure of thestability of the deformed state.

The thermal results show that for increasing stored energy, the softening processtakes place at lower temperatures. An influence of the deformation temperature be-comes apparent. Higher deformation temperatures promote easier reaction for the samestored energy. Exact analysis of these facts shows that the form function makes only anegligible contribution here. The effect is induced by a reduced activation energy.

Different types of deformation show a stronger influence on the reaction tempera-ture than the deformation temperature. At lower energies, distorted samples soften fast-er than extended or rolled ones. At higher energies, the reverse is true: Rolled samplesreact faster. It is noticeable that cyclically deformed samples, for torsion as well as forpush-pull, do not diverge from the unidirectionally deformed samples of the same de-formation mode. This is remarkable because, particularly for tension and push-pull de-formation, there are substantial differences in the activation energy.

The activation energy describes the purely temperature dependence of the reac-tion. For small deformation and stored energies of distorted copper at a value of170 kJ/mol, it lies below the activation energy of volume self diffusion (200 kJ/mol).Unidirectionally extended samples show a higher activation energy (190 kJ/mol); push-pull deformed samples, on the other hand, show significantly lower activation energies(130 kJ/mol). With increasing energy, the activation energies of all deformation typesfall. Figure 1.10 demonstrates these relationships.

With the aid of torsional deformation, it is unequivocally proved that only uponreaching the strain-hardening stage V, one may presume constant activation energy. At


8

Table 1.1: The constant k1 according to Equation (2) for various temperatures. The constant k2applies to extreme deformation in the region V.

k1 �c/G k2

Cu 253 K 6.4·10–4 1.5·10–3 4.8·10–4

Cu 293 K 5.7·10–4 1.25·10–3 4.5·10–4

Cu 373 K 5.5·10–4 1.12·10–3 4.5·10–4

Cu 293 K tension 5.0·10–4 1.5·10–3 –Ag 253 K 6.0·10–4 1.5·10–3 4.5·10–4

Ag 293 K 5.5·10–4 1.25·10–3 4.4·10–4

values of 80 to 90 kJ/mol, here for all deformation temperatures, the activation energylies in the region of grain boundary self diffusion or diffusion in dislocation cores. Ten-sion and push-pull samples do not achieve these high stored energies; for these defor-mation modes, there is therefore no region of constant activation energy. Elevated de-formation temperatures result in a lower softening activation energy. One may interpretthis as strain hardening at higher temperature producing a microstructure that softensfaster. This effect should be accounted for when setting up constitutive equations.

There is a theory for the softening of deformed metals through the mechanism ofprimary recrystallization by Johnson and Mehl [13], Avrami [14–16] and Kolmogorov[17]. In the following, this will be denoted the JMAK theory. Comparison of the mea-sured activation energies with those predicted by the JMAK theory allow conclusionsto be drawn regarding the basic mechanisms of primary recrystallization.

Accordingly, for high deformation continuous nucleation must be assumed,whereas for low deformation site, saturated nucleation is more probable. Table 1.2shows the comparison in detail. For high deformation, this interpretation complies withstudies according to the microstructural-path method [18]. The grain spectra of weaklydeformed and recrystallized material show agreement with calculated spectra after site-saturated cluster nucleation.

1.2 Experiments

9

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

Table 1.2: Effective activation energies from the JMAK theory compared with measured valuesfor low/high deformation.

Site-saturated Continuous Measured values [kJ/mol]nucleation nucleation[kJ/mol] [kJ/mol] low deformation high deformation

Copper 166–120 125–86 170 ±8 85 ±5Silver 143–115 107–86 120 ±8 85 ±5

The second component of the kinetics, the pure reaction form, is described by thefunction f �X�. For all nucleation-nucleation growth reactions, this function, by way ofthe transformed fraction, is parabolic with zero points at the beginning and end of thereaction. A more significant picture results when this function is compared with theJMAK theory. For ideal nucleation-nucleation growth reactions, the theory demands forf �X��1 � X� a higher order function of ln�1 � X� with an exponent �n� 1��n inde-pendent of X. The Avrami exponent n takes the value 4 or 3, respectively.

In reality, however, independent of the measurement method, one finds Avramiexponents that decrease with X. The thermal data show this particularly clearly. As anexample, Figure 1.11 shows the curve of the Avrami exponent as a function of thetransformed fraction for distorted copper. The horizontal reference line outlines thecurve for low degrees of deformation �� 0�8 or 1�4�, the central reference line appliesto intermediate degrees of deformation �� 2�4 or 3�0��

Rolling and cyclic torsion act in the same way as unidirectional torsion if thestored energy is taken as the comparative measure instead of the strain-hardening re-gions. Complementary studies using the transmission electron microscope show that themicrostructural details are similar for these deformations (cf. Nix et al. [9]). The defor-mation types unidirectional tension and push-pull are very different from torsion. TheAvrami exponents are very large for unidirectional tension.

In summary, the combination of stored energy, softening temperature and activa-tion energy as well as the softening form function is unequivocal for the material statesstudied here. The degree and type of deformation of a sample may thus be identifiedwith no knowledge of its prior mechanical history.


10

Figure 1.11: Avrami exponent versus the transformed fraction for distorted copper with shearstrains 3.4 ≤�≤7.0.

1.3 Simulation

Primary recrystallization as one of the main processes of thermal softening was simulatedby a cellular automaton (CA). These latter are networks of computational units, whichdevelop their properties through the interaction of numerous similar particles [19, 20].They are comprehensively described by the four properties geometry, environment, statesand rules of evolution. Cellular automatons were first applied to primary recrystallizationfor the two-dimensional case by Hesselbarth et al. [21, 22]. For the extension to threedimensions, a cubic lattice of identical cubes is defined. Each of these small cubes repre-sents a real sample volume of about 0.6 �m3. This value is obtained by comparison withreal grain sizes. The whole field is then equivalent to a mass of 0.007 mg. Compared withthe mass of thermal samples at 150 mg, this is very little. The geometrically closest cellsare counted as the nearest neighbours. It turns out that an alternating sequence of 7 and 19nearest neighbours yields the best results. Stochastically changing environments influencethe kinetics in consequence of the resultant rough surface of the growing grains.

Figure 1.12 shows the 7 nearest neighbours on the left and the 19 on the right,starting with a nucleus in the second time-step. The change of environment with eachtime-step causes all grains in odd time-steps to be identical. The resultant grain shapelooks like a flattened octahedron.

1.3 Simulation

11

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

The possible states of the cells are recrystallized and non-recrystallized. For theextension to different grain boundary velocities, the non-recrystallized state was subdi-vided further. The fourth descriptive characteristic after the geometry, environment andpossible states are the rules of evolution. These stipulate, which states the cells willadopt in the next time-step. If a cell already has a recrystallized environment, the rulespredict that in the next time-step, this cell will also adopt the recrystallized state. Usingthis simple cellular automaton, it is possible to solve the differential equation of theJMAK theory. The quality of the solution improves with the field size.

Alternatively, several calculations may be combined. The deviation of simulated fromtheoretical kinetics is of the order of 1%. A great advantage of cellular automatons is thatboundary conditions are automatically taken into account. They do not have to be statedexplicitly. This advantage should not be underestimated because the problem of collision ofgrowing grains for arbitrary site-dependent nucleation is non-trivial. In this way, it is pos-sible to calculate even complicated geometries not amenable to analytical solution.

The objective of simulations is to support the discussion on the various possiblecauses for the deviation of real recrystallization kinetics from the theoretically predictedprocesses. In so doing, one differentiates between topological and energetic causes.Namely, the classical JMAK theory leans on two hypotheses, which strongly limit its uni-versal applicability. The first in the assumption that all processes are statistically distrib-uted in space; this applies to nucleation in the first instance and thus subsequent graingrowth. Any kind of nucleation concentration on chosen structural inhomogeneities altersthe collision course of growing nuclei and hence the correction factors of the extended-volume model. The second restrictive assumption concerns the process rates. Nucleationand nuclear growth are assumed to be site-independent and constant in time. However,comparison of various strongly deformed samples shows at once that for different storedenergies, even if they are mean values, recrystallization occurs at different rates. If, there-fore, we have structural components with different energies side by side in the same sam-ple, one must be aware that a uniform process rate does not exist.

Non-statistical nucleation was intensively studied for point clustering. The modelpostulates stochastically placed centres, which show an increased nucleation rate. Thenucleation density follows a Normal distribution around the chosen centres. On a linebetween two concentration centres, one obtains the distribution for the nucleation ratesshown in Figure 1.13.

This yields two boundary cases, which are also being discussed in the literature[23–25]. First, we have very broad scatter of nuclei and, secondly, a high concentrationon the chosen sites. In a narrow parameter range between these boundary cases, the ki-netics are very sensitive to change (Figure 1.14).

It is possible to simulate the continuously decreasing Avrami exponents of thestrongly deformed samples as well as the low Avrami exponents at the beginning ofthe transformation found for weakly deformed samples.

Another structural characteristic, the contiguity, describes the cohesion of recrys-tallized areas. This quantity may also be calculated using the cellular automaton forvarious site functions of nucleation. Comparison with experimentally determined conti-guity curves indicates that nucleation clustering can also be found in real materials.The evaluation of grain-size distributions also points to clustering.


12

Introduction of site-dependent process rates is effected through an extension of pos-sible non-recrystallized states. One differentiates between mobility and driving force. Withreference to the literature [26, 27], a value around the factor 3 is taken. The result is 9different velocities. Two degrees of recrystallization are defined, one of which refers tothe energy, the other to the volume. If the kinetics of the JMAK theory are appropriatelyevaluated, there is hardly any difference between these two definitions.

The introduction of different velocities causes the reaction rate to decline towardsthe end of the transformation. If the proportionality of the areas of equal velocity andthe resulting grain size is changed, the kinetics may be influenced to a degree. The ki-netics of strongly deformed samples may be simulated if the areas of equal velocity arelarger 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 de-

1.3 Simulation

13

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

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

formed samples with low Avrami exponents cannot be calculated using this ansatz. Anexperimental indication of rate retardation is obtained from studies on strongly rolledcopper by the microstructural-path method [18].

Finally, it may be said that there are indicators for each ansatz in real recrystalli-zation processes. Considering the experimental results, a weighted mixture of bothwould 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 thestored 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 compositemay then be determined from this using a parallel or series network. In future, this typeof model coupling will become more important in those areas, where modelling withconstitutive equations on the basis of discontinuous phenomena only such as dynamicrecrystallization 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 frommechanical data, which correspond to different structures of the strain-hardenedmaterial, may also be determined from the thermally measured stored energy andfrom the rate of energy storage. One finds that the energy storage of more fullycondensed states is particularly effective.

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

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

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


14

References

[1] J. Gil Sevillano, P. van Houtte, E. Aernoudt: Deutung der Schertexturen mit Hilfe der Tay-loranalyse. 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. Ma-

ter. 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. Me-

tall. 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. Ungar, L.S. Toth, J. Illy, I. Kovacs: Dislocation structure and work hardening in poly-

crystalline of hc copper rods deformed by torsion and tension. Acta metall. 34 (1986)1257.

[11] R. Gerdes: Ein stochastisches Werkstoffmodell fur das inelastische Materialverhalten metal-lischer Werkstoffe im Hoch- und Tieftemperaturbereich. Mechanik-Zentrum der TU Braun-schweig (Dissertation), Braunschweig, 1995.

[12] H. Schlums, E.A. Steck: Description of cyclic deformation processes with a stochasticmodel 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 distri-

bution 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 Kristallvorgange in Metallen (russ. mit deutscher Zu-

sammenfassung). Akad. Nauk. SSSR Ser. Mat. 1 (1937) 335.[18] R. A. Vandermeer, D. Juul Jensen: Quantifying recrystallization nucleation and growth ki-

netics 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. Gobel: 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. Zack-

ay, 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.

References

15

[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. Tolg: Stored energy and recrystallization temperature of rolledcopper and silver single crystals with defined solute contents. Acta metall. 27 (1979) 1539.

[28] L. Kaps: Einfluss der mechanischen Vorgeschichte auf die primare Rekristallisation. ShakerVerlag, Aachen, 1997.


16

2 Material State after Uni- and Biaxial CyclicDeformation

Walter Gieseke, K. Roger Hillert and Gunter Lange*

2.1 Introduction

Technical components and structures today are increasingly being designed and dis-played by computer-aided methods. High speed computers permit the use of mathemati-cal models able to numerically reconstruct material behaviour even in the course ofcomplex loading procedures.

In phenomenological continuum mechanics, the cyclic hardening and softeningbehaviour as well as the Bauschinger effect are described by yield surface models. If aphysical microstructural formulation is chosen as a basis for these models, then it is vi-tally important to have exact knowledge of the processes occurring in the metal latticeduring deformation. Two surface models, going back to a development by Dafalias andPopov [1–4], describe the displacement of the elastic deformation zone in a dual axisstress 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 ofsmall offset strains under load become characteristically deformed. In the present sub-project, the effect of cyclic deformation on the shape and position of the yield surfacesis studied, and their relation to the dislocation structure. To this end, the yield surfacesof three materials with different slip behaviour were measured after prior uni- or biaxialdeformation. The influence of the dislocation structures produced and the effect of in-ner stresses are discussed.

17

* Technische Universitat Braunschweig, Institut fur Werkstoffe, Langer Kamp 8,D-38106 Braunschweig, Germany



2.2 Experiments and Measurement Methods

Copper of 99.99% purity was chosen as a material exhibiting typical wavy slip behav-iour. Most of the experiments were performed using the technically important materialAlMg3 of 99.88% purity. Its behaviour may be described as being somewhere inter-mediate between planar and wavy slip 1. Commercial austenitic steel 1.4404 (AISI316L) was used as a material with typical planar slip behaviour. The total strain ampli-tude 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 recrystal-lized. AlMg3 and the austenitic steel were quenched in water, the copper samplescooled in the oven. After the thermal treatment, a �100� slightly fibrous texture wasidentified, which did not change during the subsequent cyclic deformation. The coppershowed almost no texture. The yield surfaces of the initial materials were isotropic, in-dependent of the offset used [7, 8].

Tubular samples were used in the experiments. Their outer diameter and wallthickness 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 followingcyclic 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/compres-sion and alternate torsion; a 90 � antiphase combination of tension/compression and al-ternate torsion. The dislocation structures were subsequently investigated using a Phi-lips 120 kV transmission electron microscope. For the strain-controlled experiments, atriangular nominal value signal with constant strain rate of 2 ·10–3 s–1 was chosen. Theequivalent strains were calculated after v. Mises according to:

�eq � �2 � 13�2

� �1�2

with � � 1��3

� � � �1�

Two methods were applied to determine the yield surfaces. Using the definition via anoffset strain of 2 ·10–4%, the load was increased in steps of 6 N/mm2 in the �-directionor of 2.5 N/mm2 in the �-direction until the given yield limit was reached. There was a10 s intermission at each level. Before the next point on the curve was measured, sev-eral load cycles were run through again to set the material to the same initial state.

The second measurement method was the recording of directionally dependentstress-strain diagrams. Here, a new sample was used for each point measured. It wasstressed under predetermined load paths immediately following the cyclic treatment farinto the plastic region. In this way, static strain ageing effects were avoided. Further, itwas possible to determine yield surfaces of higher offset strain and areas of equal tan-gent 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 withinthe scope of this project was used:

2 Material State after Uni- and Biaxial Cyclic Deformation

18

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

�eq � �� A�2 � EG�� A�2

� �1�2

� �2�

�eq � �� A�2 � GE�� A�2

� �1�2

� �3�

The advantage of these equations lies in the fact that all equivalent stress-strain dia-grams show a Young’s modulus appropriate increase in the elastic region. In the caseof AlMg3, the �,�-hysteresis can be converted into the equivalent �eq, �eq-hysteresis,which are in almost complete agreement with the measured �, �-hysteresis values. Fig-ure 2.1 a shows the strain paths for the measurement of a family of yield surfaces ofvarying 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 re-gion after load reversal in the load maximum. Figure 2.1b shows the appropriate loadpaths, Figure 2.1 c the relevant equivalent stress-strain diagrams. The yield points ofvarious offset strains were determined by parallel shift of the elastic straight line. Forareas with the same tangent modules, the equivalent stress-strain curves were differen-tiated; for a given tangential gradient, one obtains the pertinent �,�-points.

The yield surfaces in Figure 2.2 show that the yield conditions according to Equa-tions (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, offsetstrain �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 numberof cycles for the appropriate given equivalent total strain amplitude of ��eq � �0.5%.The three proportional loads are compared and that for the 90 � anti-phase combinationof tension/compression and alternating torsion.

For all four load types, the saturation state is reached after about 500 cycles. Thecurves for proportional loading almost coincide. Larger torsional fractions cause aslight increase in the stress amplitudes. The curve for disproportional loading systemati-cally assumes higher values. This additional hardening effect is much more pronouncedat the beginning of the fatigue at about 25% than in the saturation stage, where it isonly about 5%.

Figure 2.3b shows the appropriate curves for the lower total strain amplitude of��eq � �0.3%.

2.3 Results

19


20

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equi

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ress

-str

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(3).

The saturation state is reached after about 900 cycles. Here too, the curves of pro-portional loading approximately coincide. For disproportional loading, a weak addi-tional hardening effect appears at the beginning of the fatique stage, yet this reverses insaturation.

The additional hardening effect may usually be explained by the fact that for anappropriately large plastic strain amplitude, the anti-phase loading leads to an addi-tional hardening because more slip systems are activated than for proportional loading.This is particularly the case for the high strain amplitude of ��eq � �0.5% at the be-ginning of the fatigue. For strain amplitudes of ��eq � �0.3%, the plastic fraction of

2.3 Results

21

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 ��eq � �0.5%, material: AlMg3.

saturation is so small that the additional hardening in consequence of anti-phase load-ing is not enough to compensate the overall smaller stress values of the sum of the in-dividual components. At strain amplitudes of ��eq � �0.4%, AlMg3 shows Masing be-haviour for all proportional loads. Deviations occur at smaller amplitudes: The lengthof the elastic regions increases with decreasing strain amplitude. Similar behaviour isfound for planar flowing �-brass [10].

For copper, a total strain amplitude of ��eq � �0.5% under phase-shifted loadingproduces a pronounced additional hardening effect throughout the whole fatigue region(Figure 2.4 a).

As for AlMg3, the curves for proportional loading approximately coincide, thoughthe pure torsional load yields the lowest values. The stress values of the phase-shifted


22

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

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

loading reach saturation after about 30 cycles. Under proportional loading, on the otherhand, constant stress values are only measured after about 50 cycles. For an amplitudeof ��eq � �0.1%, the effect occurs only at the onset of fatigue (Figure 2.4b).

For a further reduction to ��eq � �0.05%, the stress values for phase-shiftedloading in the saturation region lie below those for synchronous loading (Figure 2.4 c).

The considerations regarding the additional hardening effect in AlMg3 are equallyapplicable here.

The austenitic steel 1.4404 for proportional loading at ��eq � �0.75% shows a re-latively short strain hardening region already reaching saturation after about 20 cycles. Butthe 90 � phase-shifted loading produces a strong additional hardening effect. The appropri-

2.3 Results

23

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

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

ate stress values compared with proportional loading are increased by more than 60%. Thesaturation plateau is only reached after about 30 cycles (Figure 2.5 a).

For a strain amplitude of ��eq � �0.5% too, the material reaches saturation forproportional loading after about 20 cycles. The increase of the stress amplitudes is lesshere, however. Phase-shifted loading (Figure 2.5 b) also yields a distinct additionalhardening effect. The appropriate stress amplitudes as for ��eq � �0.75% are greatlyincreased. The additional hardening effect may be regarded here as a consequence ofthe 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 typesstudied, the dipolar walls lie in (111) planes at the onset of fatigue. The value and type


24

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

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

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 ��eq � �0.3%) after proportional and disproportionalloading, the (110) walls and the initial (111) walls dominate. In almost all cases, forma-tion of the (110) walls was the work of a single slip system. Hereby, the walls were com-pressed perpendicular to the Burgers vector. Similar structures are also found in brass with15 at% zinc [13]. At high amplitudes ��eq � �0.3%) after proportional loading, the(100) besides the (311), (210), (211) and (110) walls are predominant. By contrast, forphase-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, usuallyactivates more than four slip systems. Figure 2.6 a shows the typical example of a dis-location structure after proportional loading. The arrangement can be designated aniso-tropic since the dipolar walls in almost all grains are oriented in only one or two crys-tallographic directions. The anisotropy essentially results from the small number of ac-tive slip systems in the proportional loading case, expressing a certain planarity in theslip behaviour.

Disproportional loading at high total strain amplitudes, however, results in generallymore isotropic structures (Figure 2.6 b). Here, the dipolar wall structure is quite often de-stroyed along favourably oriented (111) planes (Figure 2.6 c). Parallel arrays of elongatedscrew dislocations are often observed in these bands, which infers high local slip activity.

Depending upon loading amplitude, for copper, characteristic dislocation struc-tures evolve, which differ much more strongly from each other than for AlMg3. In satu-ration, copper does not show Masing behaviour. The saturation state, depending on am-plitude, is reached following various amounts of accumulated plastic strain. On the ba-sis of the experimental results, it appears meaningful to classify into small(��eq � �0.2%), medium (�0.2%� ��eq � �1%) and high (��eq � �1%) ampli-tudes. After Hancock and Grosskreutz [14], in the medium amplitude region(��eq � �0.375%) at the onset of fatigue, bundles of multipoles initially appear sepa-rated by dislocation poor regions. The majority of dislocations in the bundles are pri-mary edge dislocations in parallel slip planes, which mutually interact in some sectionsto form dipoles and multipoles. Further, as for AlMg3, prismatic loops are formedthrough jog-dragging processes. Screw dislocations on the other hand are hardly foundin 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 secondarydislocations increases in the bundles. The dipoles are divided into small pieces throughcutting processes with dislocations of other slip systems. This causes additional harden-ing: The dipole ends now present in higher concentrations are less mobile. A similarprocess is also presumed for AlMg3. The bundles gradually combine to cell-like struc-tures. Finally, elongated dislocation cells are produced, the walls of which are sharplyoutlined against the dislocation poor interstices. The walls comprise short dipoles ofhigh density. In the dislocation poor regions, screw dislocations stretch from one wallto the next (Figure 2.7 a, proportional loading with ��eq � �0.5%). According to Lairdet al. [15], one may expect the spatial arrangement of the structure in Figure 2.7 a toyield approximately cylindrical dislocation cells, the cross-sectional areas of which areshown here.

2.3 Results

25

After 90 � phase-shifted overlap of tension/compression and alternate torsion, in cop-per with an equivalent strain amplitude of ��eq � �0.5%, isotropic cells dominate. Theirwalls are composed of elongated, regularly ordered single dislocations (Figure 2.7 b) asfound by Feltner and Laird for the high plastic strain amplitude ��pl � �0.5% [16].


26

a) b)

c)

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

The lack of dipolar structures is explained by Feltner and Laird as being due tounhindered cross-slip. The rotating stress vector activates the slip systems required tocreate isotropic cell structures at even smaller stress amplitudes than in the proportionalcase. Annihilation of screw dislocations is facilitated, thus producing the dislocationpoor inner cell regions. In addition, the enhanced cross slip ability of the screw disloca-tions suppresses the creation of prismatic loops.

Thus copper, for proportional and disproportional loading at ��eq � �0.5%, al-ways exhibits different slip mechanisms. For proportional loading, the screw disloca-tions glide to and fro parallel to the walls in the dislocation poor areas. At the sametime, new screw dislocations are continually being pressed out of the walls until theyreach 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 reincorporatedinto the opposite cell wall. It follows that copper shows an additional hardening effect,which is retained in saturation (cf. Figure 2.4a). For austenitic steel 1.4404, the addi-tional hardening effect predominates at 90 � phase-shifted loading with equivalent totalstrain amplitude of ��eq � �0.75% and �0.5%. Study using the transmission electronmicroscope shows for disproportional loading that although a large number of stackingfaults are produced, there is no deformation-induced martensite. For steel 1.4306, thistransformation already occurs at strain amplitudes of ��pl � �0.3% under uniaxialloading [17]. Figure 2.8 a shows a typical dislocation structure after proportional load-ing with ��eq � �0.75%. The walls of the elongated cells comprise dislocation bun-

2.3 Results

27

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

a) b)

dles with a preferential orientation parallel to the {111} planes. On the other hand, thestronger tendency to multiple slip produces a labyrinthine structure after phase-shiftedloading (Figure 2.8b). 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 withequivalent strain amplitude of ��eq � �0.5%. The materials were in the cyclic satura-tion state.

2.3.3.1 Yield surfaces on AlMg3

Figure 2.9 collates the dynamically measured 0.01% offset yield surfaces for AlMg3 forthe four chosen loading types. The starting point each time was the reversal point ofthe stress hysteresis.

The yield surfaces for proportional loading (in the following denoted proportionalyield 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 dis-proportional yield surfaces) come closest to an isotropic shape (v. Mises ellipse). The


28

a) b)Figure 2.8: a) Dislocation structure after proportional loading, ��eq � �0.75%, saturation; b) dis-location structure after 90 � phase-shifted loading, ��eq � �0.75%, saturation.

proportional yield surfaces, by contrast, show definitely anisotropic shapes. Comparedwith the axial ratio of the v. Mises ellipse, the values measured perpendicular to theloading direction (transverse yield surface values) are clearly larger than the cross sec-tions found in the loading direction (longitudinal yield surface values). As the compari-son of yield surfaces measured at the upper reversal point (Figure 2.9) and at the stresszero crossover (cf. Figure 2.2) shows, both the transverse values and the contractedlongitudinal 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 essentiallysymmetrical ellipse in the stress zero crossover. During the further course of the nega-tive half-cycle, this then changes into a flattened shape once more (flattening again onthe origin side). This deformation may also be observed on yield surfaces with thesmall offset strain of 2 ·10–4% and on tangent module areas with high tangential gradi-ents.

Figure 2.10 shows the proportional and disproportional yield surfaces measured atthe load reversal points for the relatively large offset strain of 0.2%.

In consequence of the high plastic fraction, during deformation, all four yield sur-faces practically coincide and are almost elliptical in shape. Referring to the v. Misescondition or Equation (2), the longitudinal values are slightly less than the transverseones. The surfaces thus show, in weaker form, the same anisotropy as those measuredwith small offsets. The torsional yield surface is slightly flattened in the relief direction.

2.3 Results

29

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

2.3.3.2 Yield surfaces on copper

Copper behaves in many aspects like AlMg3. In Figure 2.11, the three proportionalyield surfaces measured at the load reversal points are shown in contrast with the dis-portional surface.

The proportional surfaces are again flattened in the relief direction. The shortenedaxis too remains the same throughout the whole hysteresis cycle. The disproportionalyield surface approaches the elliptical shape, which is significantly larger. The distinctadditional 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 superposi-tion 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 flat-tening on the origin side are clearly seen.

Figure 2.13 represents the 0.02% offset yield surfaces measured after disproportionalloading at the reversal points of tension and compression (�=0). For this load path, theyield surface follows the rotating stress vector. Both yield surfaces are symmetrical tothe tensile stress axis and again show the typical flattening on the origin side.


30

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

2.4 Sequence Effects

On AlMg3 and copper in the saturation state, the variation of the loading directionfrom tension/compression to alternating torsion, and the reverse, was investigated. Theexperiments were meant to show how inner stresses affect the shape of the yield sur-faces. For the offset strain of 0.01%, the points of yield onset were taken from the as-


31

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

cending and descending branches of the hysteresis curves and from these, the yield sur-faces’ diameters (cross sections) determined. In similar fashion, the diameters of thesurfaces with equal tangent modules were also determined [9, 18]. In addition, the tran-sition of the maximum stress amplitude to the new saturation state was observed. Fig-ure 2.14 shows the change of the 0.01% yield surface diameter of AlMg3 and copper(equivalent total strain amplitude ��eq � �0.5%) for the transition from pure tension/compression to pure alternating torsion.

The broken lines show the transverse values of the tension/compression saturationyield surface, respectively (state before the change). The continuous lines represent thelongitudinal values of the saturation yield surfaces, which would have appeared follow-ing pure torsion. The yield surfaces’ diameters of torsion hysteresis in the case ofAlMg3 already decrease drastically in the first cycle and quickly reach a new saturationstate 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 in-ner stress, or else, in consequence of isotropic hardening, the saturation yield surfacesare larger after prior tension/compression than after pure alternating torsion.

The second option is confirmed by the dislocation structure. As already demon-strated, for AlMg3, an anisotropic dislocation structure evolves after proportional load-ing. In extensive grain areas, only few slip systems are activated; the dipolar walls gen-erally take up only one or two crystallographic directions. Since different slip systemsare involved in tension/compression loading than in alternating torsion, the dipolarwalls orient themselves in different crystallographic directions. The screw dislocationsmove in dislocation poor channels parallel to the dipolar walls, adjacent to the respec-tive slip systems. If a tension/compression experiment is immediately followed by onewith alternating torsion, the dislocation structure is initially unfavourable for torsion.With changing loading direction, the sources of torsional slip dislocations are activatedfirst and then later on, the dipolar walls change their orientation to one more favour-able for torsional loading. As is seen from Figure 2.14, the greater fraction of the tor-sional slip dislocations is already activated in the first three cycles after the change of


32

Figure 2.13: 0.02% offset yield surfaces measured at the load reversal points of the tension/com-pression hysteresis, disproportional loading, saturation, steel 1.4404, ��eq � �0.5%.

the loading direction. The result is an immediate drastic reduction of the yield surfacediameter. However, further restructuring of the dislocation arrangement appears to beimpeded; the diameter remains constant during subsequent cycles. The dislocationstructure anticipated for pure torsional loading is clearly unable to evolve followingprevious tension/compression loading. The large number of activated slip systems afterthe change in loading direction may offer some explanation.

For copper, the yield surface diameter from torsional hysteresis also seriously de-creases in the first torsional half-cycle after changing the loading type. Moreover, incontrast to AlMg3, it falls continuously until the saturation longitudinal value for puretorsion is reached. The longitudinal values for saturation yield surfaces thus comeabout independent of previous history. The same is true for the diameters of the tangentmodule areas. The dislocations in copper arrange themselves in a similarly isotropicway as in AlMg3 (elongated cells, dipolar walls: see Figure 2.7 a). Yet after the changein loading direction, they reorientate themselves completely. This property characterizesmaterials with wavy slip behaviour [15].

In a further experiment to assess the effect of inner stress, samples of AlMg3 andcopper were relieved from various points in the torsional hysteresis branch. The yieldsurface diameters were taken from these partial cycles and plotted in Figure 2.15 as afunction of the offset strain and of the strain values (initial stress relieving points).


33

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

For a given offset strain, both materials showed the same yield surface longitudi-nal values at every point in the hysteresis. The influence of inner stress, assumed loaddependent, upon the shape of the yield surfaces would therefore not appear to be signif-icant. 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 (AISI316L) after tension/compression and alternating torsional loading as well as propor-tional and phase-shifted superposition of both loads. The materials were first cycled tosaturation with maximum deformation amplitudes of �0.75%, whereby substantial ad-ditional hardening effects occurred. The development of the appropriate dislocationstructures was studied using a transmission electron microscope.


34

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

Yield surfaces measured in all three materials at the reversal points of the stressdeformation hysteresis, for small offset strains (0.01% or 2 ·10–4%), after proportionalalternate loading, show a flattened shape in the off-load direction compared with thev. Mises ellipse. At the stress zero crossover points of the hysteresis, the yield surfacesassume a symmetrical shape. Transverse and longitudinal values of the yield surfacesremain constant independent of the starting point in the hysteresis. This behaviour andthe sequence effects confirm that the anisotropy of the yield surfaces is caused by theappropriately anisotropic dislocation structure of the materials. Inner stresses obviouslyplay a minor role.

After disproportional loading, generally isotropic yield surfaces result. This maybe explained quite simply by the relevant isotropic dislocation structures. Yield surfacesof higher offset strains and areas of equal tangent modules for small tangential gradi-ents also evolve essentially isotropically since sufficient slip systems are activated dur-ing 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 theexperimental 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. Journalof 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 Experi-ments. Journal of Applied Mechanics 85 (1986) 298–308.

[4] F. Ellyin: An isotropic hardening rule for elastoplastic solids based on experimental obser-vations. Journal of Applied Mechanics 56 (1969) 499.

[5] N.K. Gupta, H.A. Lauert: A study of yield surface upon reversal of loading under biaxialstress. Zeitschrift fur 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 alu-minium. Journal of Strain Analysis 6(4) (1971) 263.

[7] R. Hillert: Austenitische Stahle bei ein- und bei zweiachsiger, plastischer Wechselbeanspru-chung. Dissertation TU Braunschweig, 2000.

[8] W. Gieseke, G. Lange: Veranderung des Werkstoffzustandes bei mehrachsiger plastischerWechselbeanspruchung. In SFB Nr. 319 Arbeitsbericht 1991–1993, TU Braunschweig.

References

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[9] W. Gieseke: Fließflachen und Versetzungsstrukturen metallischer Werkstoffe nach plas-tischer Wechselbeanspruchung. Dissertation TU Braunschweig, 1995.

[10] H. J. Christ: Wechselverformung der Metalle. In: B. Ilschner (Ed.): WFT Werkstoff-For-schung und Technik 9, Springer Verlag Berlin, 1991.

[11] J. I. Dickson, J. Boutin, G.L. ’Esperance: An explanation of labyrinth walls in fatiguedf.c.c. metals. Acta Metallurgica 34(8) (1986) 1505–1514.

[12] J.L. Dickson, L. Handfield, G.L. ’Esperance: Geometrical factors influencing the orienta-tions of dipolar dislocation structures produced by cyclic deformation of FCC metals.Materials Science and Engineering 81 (1986) 477–492.

[13] P. Lukas, M. Klesnil: Physics Status solidi 37 (1970) 833.[14] J.R. Hancock, J.C. Grosskreutz: Mechanisms of fatigue hardening in copper single crys-

tals. 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. Disloca-

tion structures and mechanism. Acta Metallurgica 15 (1967) 1633–1653.[17] M. Bayerlein, H.-J. Christ, H. Mughrabi: Plasticity-induced martensitic transformation dur-

ing cyclic deformation of AISI 304L stainless steel. Materials Science and Engineering A114 (1989) L11–L16.

[18] W. Gieseke, G. Lange: Yield surfaces and dislocation structures of Al-3Mg and copperafter biaxial cyclic loadings. In: A. Pineau, G. Cailletaud, T. C. Lindley (Eds.): Multiaxialfatigue and design, ESIS 21, Mechanical Engineering Publications, London, 1996, pp. 61–74.


36

3 Plasticity of Metals and Life Prediction in the Rangeof Low-Cycle Fatigue: Description of DeformationBehaviour and Creep-Fatigue Interaction

Kyong-Tschong Rie, Henrik Wittke and Jurgen Olfe*

Abstract

Results of low-cycle fatigue tests are presented and discussed, which were performed atthe Institut fur Oberflachentechnik und plasmatechnische Werkstoffentwicklung of theTechnische Universitat Braunschweig, Germany. The cyclic deformation behaviour wasinvestigated at room temperature and high temperatures. The investigated materials arecopper, 2.25Cr-1Mo steel, 304L and 12%Cr-Mo-V steel. (Report of the projects A5and B4 within the Collaborative Research Centre (SFB 319) of the Deutsche For-schungsgemeinschaft.)

3.1 Introduction

Low-cycle fatigue (LCF) and elasto-plastic cyclic behaviour of metals represent a con-siderable interest in the field of engineering since repeated cyclic loading with high am-plitude limit the useful life of many components such as hot working tools, chemicalplants, power plants and turbines. During loading in many cases after a quite smallnumber of cycles with cyclic hardening or softening, a state of cyclic saturation isreached. This saturation state can be characterized by a closed stress-strain hysteresis-loop. Cyclic deformation in the regime of low-cycle fatigue (LCF) leads to the forma-tion of cracks, which can subsequently grow until failure of a component part takesplace.

The crack growth is correlated with parameters of fracture mechanics, which takeinto account informations especially about teh steady-state stress-strain hysteresis-loops.

37

* Technische Universitat Braunschweig, Institut fur Oberflachentechnik und plasmatische Werk-stoffentwicklung, Bienroder Weg 53, D-38106 Braunschweig, Germany



Therefore, a more exact life prediction is possible by investigating the cyclic deforma-tion behaviour in detail and describing the cyclic plasticity, e.g. with constitutive equa-tions. In this paper (see Section 3.3), the investigated cyclic deformation behaviour wasdescribed by analytical relations and, moreover, by relations, which take into accountphysical processes as the development of dislocation structures.

When components are loaded at high temperature, additional processes are super-imposed on the fatigue. Besides corrosion, which is not discussed here, creep deforma-tion and creep damage are the most important. Therefore in many cases, not one typeof damage prevails, but the interaction of both fatigue and creep occurs leading to fail-ure of components.

A reliable life prediction model for creep-fatigue must consider this interaction asproposed 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 grainboundary cavities. Cavities are for many steels and some other metals the typical creepdamage and also play an important role in the case of creep-fatigue. The possibility ofunstable crack advance, which is the criterium for failure, is given if a critical config-uration of the nucleated and grown cavities is reached.

Therefore, the basis for reliable life prediction is the knowledge and descriptionof the cavity formation and growth by means of constitutive equations. In the case ofdiffusion-controlled cavity growth, the distance between the voids has an important in-fluence on their growth. This occurs especially in the case of low-cycle fatigue, wherethe cavity formation plays an important role. Thus, the stochastic process of pore nu-cleation on grain boundaries and the cyclic dependence of this process have to be takeninto consideration as a theoretical description. The experimental analysis has to detectthe cavity size distribution, which is a consequence of the complex interactions be-tween the cavities (see Section 3.4.2).

Formerly, the total stress and strain have been used for the calculation of thecreep-fatigue damage. However, these are macroscopic parameters, whereas the crackgrowth is a local phenomenon. Therefore, the local conditions near the crack tip haveto be taken into consideration. The determination of the strain fields in front of cracksis 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 polycrystal-line copper and the steel 2.25Cr-1Mo (10 CrMo9 10). Specimens of 2.25Cr-1Mo wereinvestigated in as-received conditions, in the case of copper, the material was annealedat 650 �C for 1/2 h.

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

38

The tests were controlled by total strain and carried out at room temperature inair. The strain rates were ��=10–3 s–1 (or, for a small number of tests, ��= 2 · 10–3 s–1) forsteel, 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 ��/2, some tests were performed as two-steptests (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 waschanged in the next step. The strain amplitudes were in general in the low-cycle fatiguerange and a few amplitudes in the range of high-cycle fatigue (HCF) and in the transi-tion regime between low-cycle fatigue and extremely low-cycle fatigue (ELCF): Thetests with copper were performed with strain amplitudes between 0.1 and 1.7%, thetests with steel with amplitudes between 0.185 and 1.2%.

The incremental-step tests were carried out with constant strain rate and with giv-en values for the lowest and the highest strain amplitude, (��/2)min and (��/2)max. Thefactor of subsequent amplitudes qa in the ascending part of the IST-block or, alterna-tively, the difference of amplitudes �a is constant.

For most of the tests, smooth cylindrical specimens were used. Usually, the diam-eter and the length of the gauge were 14 mm and 20 mm, for the tests with very highstrain amplitudes (near the ELCF-regime), the diameter was 14.7 mm and the length10 mm. For some tests, flat specimens were used with the values 8.7 ×5 mm2 for therectangular cross-section.

The steady-state microstructure of tested specimens was investigated with trans-mission electron microscope at the Institut fur Schweißtechnik (Prof. Wohlfahrt [1]),the Institut fur Metallphysik und Nukleare Festkorperphysik (Prof. Neuhauser [2]) andthe Institut fur Werkstoffe (Prof. Lange [3]). They are all at the Technische UniversitatBraunschweig 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 witha tension hold time up to 1 h at 600 � and 650 �C for the 304L, and 550 �C for the ferri-tic steel. For the tests for the lifetime determination and the tests for analysing the cavi-ty configuration, we used round and polished specimen. After low-cycle fatigue testing,the specimens were metallographically prepared for stereological analysis of the densityand cavity size distribution (see Section 3.4.2.1).

A furnace with a window and special optics allow high magnification observationof the specimen surface continuously during the test with a video system and a subse-quent measurement of the crack growth, the crack tip opening and the crack contour onflat and polished specimens in an inert atmosphere. In-situ measurement of the strainfield 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 linedistance of 200 �m, which was photographed at the beginning of the test and at givenloads after cycling. By means of digital image analysis, the local strain at every crossof the grating was calculated by the group of Prof. Ritter [10] and Dr. Andresen [11]

3.2 Experimental Details

39

(TU Braunschweig, Collaborative Research Centre (SFB319)). 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 illustratedby means of a straight line, and the line, which surrounds the 5% deformed zone at thecrack tip, is shown in the figure. From the figure, the size of the 5% deformed zone indirection of its maximum expansion was taken. In the following, this distance wasdesigned as R0.05 in analogy to Iino [12]. It has been used to describe the developmentof the highly deformed zone in dependence on the crack length and the tension holdtime.

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 rangeof lifetime. After a quite small number of cycles, the end of a rapid hardening regimeis reached. Due to the effect of secondary hardening, in some ranges of amplitudes, nosaturation was observed, but, as first approximation, the effect of secondary hardeningcan be neglected [13]. Examples for cyclic hardening curves up to saturation are shownin Figure 3.2 a.

In the case of single-step tests with 2.25Cr-1Mo, there is cyclic softening innearly the whole range of strain amplitudes. In the first cycles, rapid hardening can befound before cyclic softening takes place. After this, a steady-state regime can be


40

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

found, which continues until a failure takes place. While in the case of copper, there isa very clear effect of rapid hardening, in the case of 2.25Cr-1Mo, the effects of cyclichardening and softening are less pronounced.

For both materials, after a certain number of cycles, a state of saturation isachieved. The stress-strain behaviour is represented by a hysteresis-loop. (To avoid con-fusion, it may be useful to mark characteristic values of the steady-state hysteresis-loopwith an index. For example, the amplitude of stress ��/2 can be written in the case ofsaturation as (��/2)s. Nevertheless, no index is used in this paper because it is usuallyclear from the context whether the instantaneous or the steady-state values are re-ferred.)

In Figure 3.2 b, an example for cyclic stress-strain curves, ��/2 or ��i/2 vs. ��p/2,are shown, which are constructed with the aid of steady-state hysteresis-loops. The valuesof 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 be-tween the amplitudes of plastic strain ��p/2, total strain ��/2 and stress ��/2. The am-plitudes of the internal stress, ��i/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 24tests with amplitudes in the range of LCF (single-step tests and two-step tests withlow-high amplitude-sequences; 0�16� � ��2 � 1�0��. Additionally, one test in thehigh-cycle fatigue (HCF) regime and three tests in the transition regime between LCFand extremely low-cycle fatigue (ELCF: compare Komotori and Shimizu [15]) aretaken into consideration. In the case of copper, the 24 tests are used to study various

3.3 Tests at Room Temperature: Description of the Deformation Behaviour

41

Figure 3.2: Copper; a) cyclic hardening curves, ��=10–4 s–1; b) cyclic stress-strain curves: amplitudesof 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.

parameters of the material, in the case of 2.25Cr-1Mo, data are used from 13 single-step tests (12 LCF-tests, one HCF-test).

Examples for steady-state hysteresis-loops are shown in the Figure 3.3 a and b. InFigure 3.3 a, stress-strain hysteresis-loops of 2.25Cr-1Mo are shown, in Figure 3.3 b, hys-teresis-loops in relative coordinates, �r and �r, are shown in the case of copper. The relativecoordinate system is defined by an origin, which is set at the point of minimum stress andstrain of the hysteresis-loop. The material exhibits Masing behaviour when the upperbranches of different hysteresis-loops follow a common curve in the relative coordinatesystem. In contrast, copper exhibits non-Masing behaviour in single-step tests as can beseen in Figure 3.3 b. Also for 2.25Cr-1Mo, non-Masing behaviour was found. Only in asmall range of the tested amplitudes, in the range of 0.185%<��/2<0.4%, the steel ex-hibits approximately Masing behaviour.

For many materials with non-Masing behaviour, it is possible to get a “mastercurve”, which is obtained from matching the upper branches of the hysteresis-loopsthrough 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 thetested materials in good approximation (Schubert [18], Rie et al. [19]). This behaviouris 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 befound, which is equal to that of an equivalent single-step test. In the range of the testedamplitudes, this behaviour can also be found in good approximation in tests with high-low amplitude-sequences. The materials are nearly history-independent (compare Felt-ner and Laird [20] and Hoffmann et al. [21]).

Also in incremental-step tests, a state of cyclic saturation can be found. In con-trast to the stress-strain behaviour in single-step tests and two-step tests, the steady-state stress-strain behaviour in incremental-step tests can be approximately expressedby Masing behaviour (see [8, 13]).


42

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

3.3.2 Microstructural results and interpretation

For both materials, dislocation cell structures were found. For 2.25Cr-1Mo, cell structurewas found in single-step tests in the range of amplitudes, in which the materials exhibitnon-Masing behaviour. In the case of single-step tests with copper, the cell structure iswell developed for high amplitudes, for low amplitudes, other dislocation structures aredominating as e.g. vein structure. Often, the shape of the cells is not cuboidal but elon-gated. With increasing strain amplitude, the cell size is decreasing (compare Feltner andLaird [22]). Schubert [18] proposed a microstructure-dependent cyclic proportional limit

�prop � �L � 2MS Gb�dm � �2�

where �L is the lattice friction stress, MS is the Sachs factor, G is the shear modulusand b is the absolute value of the Burgers vector. The decrease of the mean cell sizedm and the increase of �prop with increasing strain amplitude is in agreement with thenon-Masing behaviour of the materials [13, 14]. For 2.25Cr-1Mo, the value of dm inEquation (2) corresponds to the mean distance of precipitates for low amplitudes and tothe mean cell size for high amplitudes (��/2 > 0.4%). Therefore, in the case of low am-plitudes, Masing behaviour was found [18].

A typical steady-state dislocation structure of the second step of a two-step testwith an amplitude-sequence high-low is shown in Figure 3.5. A dislocation cell struc-ture can be seen although the dominating structure of the low amplitude in the case ofa single-step test is vein structure (see [18]). While in two-step tests with amplitude-se-quences low-high the microstructure is history-independent, it is obviously not indepen-dent in the case of a test with an amplitude-sequence high-low (compare [21]). Never-theless, the dependence of the macroscopic behaviour on this history-dependent micro-structural behaviour is almost negligible.

In incremental-step tests with sufficiently high values of (��/2)max, dislocationcell structure can be found in cyclic saturation. The dislocation structure is assumed as


43

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

quasi-stable: In cyclic saturation, the dislocation structure does not change within oneIST-block. The quasi-stable dislocation structure correlates well with the Masing behav-iour 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 � for ��2 � 0�2� �

dm � 0�76 � for ��2 � 0�4� �

dm � 0�58 � for ��2 � 0�7� �

in the case of copper and SSTs for �� 10�4 s�1. In the case of 2.25Cr-1Mo, SSTs,�� 10�3 s�1, the experimental values are:

dm � 0�85 � for ��2 � 0�6� � and

dm � 0�65 � for ��2 � 1�2� �13� �

These values were used to calculate the cyclic proportional limit �prop, and a goodagreement 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 (Sections3.3.4.1 and 3.3.4.2).


44

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

3.3.3 Phenomenological description of the deformation behaviour

3.3.3.1 Description of cyclic hardening curve, cyclic stress-strain curveand hysteresis-loop

It is shown by Wittke [13] that the first part of a cyclic hardening curve of a single-steptest, the rapid hardening regime, can be described excellently with a stretched exponentialfunction for stress amplitude ��/2 vs. cycle number N (or more exact: N – 0.25):

��2 � A0 � �As � A0� 1 � exp � �N � 0�25��N0� ��H

� �� 3�

The constants A0 and As are closely related to the monotonous and cyclic stress-straincurve, respectively, the constants N0 and �H are found by trial and error. A simple de-pendence of the parameters on the steady-state value of the plastic portion of the totalstrain amplitude can be found [8]. Moreover, the stretched exponential function, ��/2vs. N, is applicable also for two-step tests in the case of hardening and softening ingood approximation. The comparison between experimental and calculated cyclic hard-ening 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 3.2b, in the case of copper, the description of the cyclicstress-strain curves by the solid line and the dotted line is quite good. The double-loga-rithmic cyclic stress-strain curves, ��/2 vs. ��p/2, for different strain rates are nearlyparallel. Also in the case of 2.25Cr-1Mo, the description of the cyclic stress-straincurve by a power law is good. In the case of 2.25Cr-1Mo, we get with

��2 � k ��p�2�n �4�

and by using the constants k � 803 MPa and n � 0�138 good agreement between ex-perimental and calculated values �� 10�3 s�1 � E � 208 GPa�.

For copper, the values of the constants for the different css-curves in Figure 3.2b are:

k � 554�6 MPa � n � 0�228 for �� 10�3 s�1 �

k � 565�9 MPa � n � 0�238 for �� 10�4 s�1 � 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 thesteady-state hysteresis-loop is the most important point in this Section 3.3. In first ap-proximation, also in the case of the hysteresis-loop, a power law between relative stressand relative plastic strain, �r and �pr, can be assumed (see Morrow [23]):

�r � kH ��pr � �5�

It should be mentioned that the parameters kH and � are dependent on the plastic strainamplitude.


45

Although the description of the hysteresis-loop with a power law is quite rough, itmay be useful to apply such a law for fracture mechanical estimation (see Rie andWittke [24]). To get a better description of the hysteresis-loop shape, other relations arenecessary. The hysteresis-loop can be described, e.g. by a two-tangents method (com-pare [19]), as follows:

�pr � ��r�k0�1��0 � ��r�kE�1��E � �6�

For each hysteresis-loop, four constants, k0, �0, kE and �E, have to be determined. Anexample for the applicability of this relation in the case of the mild steel Fe510 (St-52),which was tested at the Institut fur 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 areexpressed by:

�r � AG �1 � exp ��pr��G��G�� 7�

or alternatively by:

�r � Cq exp ��q �ln ��pr��E��2� � �8�

The three constants of the stretched exponential function (Equation (7)) are �G, �G and�G, the constants of the exponential parabola function (Equation (8)) are Cq, �q and �E.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 sec-ond derivative of the hysteresis-loop branch, d2�r/d�

2r vs. �r (see Section 3.3.4.1).


46

Figure 3.6: Mild steel Fe510; hysteresis-loop in relative coordinates; ��/2=0.5%, ��=10–4 s–1;comparison between experiment and calculation; calculation according to Equation (6), k0 =45418MPa, �0 =0.553, kE =1238 MPa, �E =0.095; E=210 GPa.

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 constantscan be obtained when the parameters of the power law (Equation (5)), kH and �, aregiven as simple functions of the plastic strain range ��p. Such functional relations aredeveloped in [13, 26]. Furthermore, a method to calculate the parameters of the expo-nential parabola function (Equation (8)), Cq, �q, and �E, in dependence on the plasticstrain amplitude is described in [13].

As shown above, in the case of non-Masing behaviour, it is possible to get a mas-ter curve. This master curve together with the cyclic stress-strain curve can be used toconstruct each hysteresis-loop (see [17]). In contrast to the power-law master-curve pro-posed by Lefebvre and Ellyin [17], better results were achieved, e.g. by a stretched ex-ponential function or an exponential parabola function (see [13]). In the latter case, themaster curve can be described by:

� � Cq exp ��q

�ln ��pr��E��2� � �9�

where Cq, �q

and �E 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 � �q

� 0�03576 � �E � 2�3194� �

All these methods, which were used to describe various steady-state hysteresis-loops ofcopper 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 necessaryto take into consideration that the applied stress � can be separated into the internaland the effective stress, �i and �eff. The effective stress is that fraction of the totalstress causing dislocations to move at a specific velocity, the internal stress can be de-fined as the stress needed to balance the dislocation configuration at a net zero value ofthe plastic strain rate (see Tsou and Quesnel [27]).

At room temperature, internal stress can be easily obtained experimentally bystress relaxation tests. For this purpose, test specimens were cycled to approximatedsaturation 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 inter-nal stress �i plotted vs. the plastic strain �p. In agreement with the method of Tsou andQuesnel [27], the stress value after 30 min of relaxation is adopted as the internalstress value. Figure 3.7 b shows for a stress relaxation test performed after a monotonicstrain-controlled tension test (�� 10�4 s�1� that this is a good approximation: After


47

less than 30 min=1800 s, the stress value is almost constant. (In Figure 3.7 b, t=0 isdefined by the start of the stress relaxation procedure.)

The twelve experimental points of Figure 3.7 a are expressed by the dotted-line fitcurve. 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 stress-strain curve ��i/2 vs. ��p/2 can be determined. This cyclic stress-strain curve has beenshown already in Figure 3.2b. With the same amplitude of the plastic strain, the shapeof the �i-�p-hysteresis-loop is assumed to be independent of the strain rate of the priorcyclic test (compare Tsou and Quesnel [27], Hatanaka and Ishimoto [28]).

The proposed Equations (7) and (8) are well appropriate to fit the experimentalpoints. Therefore, one of them (here Equation (8)) is used in the following for checkingwhether the �i-�p-hysteresis-loops exhibit Masing or non-Masing behaviour. To investi-gate the Masing or non-Masing behaviour of the �i-�p-hysteresis-loops, several hyster-esis-loops are presented in Figure 3.8 a in relative coordinates. In this figure, �ir-�pr-hys-teresis-loops for a small, a medium and a quite large strain amplitude of the LCF rangeare shown. Non-Masing behaviour can be seen clearly.

In the following, the dependence of the non-Masing behaviour on microstructurewill be quantified with the model of Schubert [18]. As usual, a macroscopic cyclic pro-portional limit can be defined by a strain offset, e.g. 0.01%. Nevertheless, the value ofthe 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 Equa-tion (1):

�r � �pr � �ir�E � �10�


48

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

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 deri-vative 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 vol-umes with different yield stresses (compare Polak et al. [29]): �yc is interpreted as theyield stress with the highest probability density within the material. By this definition,an uniquely applicable and physically better justified macroscopic cyclic proportionallimit is found. With the values of dm and �prop (see Equation (2)) for copper given bySchubert [18] or Rie et al. [19], respectively, the good agreement between the two cyc-lic proportional limits, �prop and �yc, can be seen in Figure 3.8b.

The effective stresses contribute also to the non-Masing behaviour of materials, butin agreement with the above mentioned model, the main reason of the non-Masing behav-iour 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 ofa stress relaxation test for another charge of the material give a value of ��i/��=0.907for ��/2=0.6%. This value is quite similar to the tests with copper.


49

Figure 3.8: Copper; a) hysteresis-loops of relative internal stress vs. relative plastic strain (withoutexperimental values; dotted line hysteresis-loops: calculation by Equation (8)); parameters of theformer performed single-step tests: strain rate ��=10–4 s–1; strain amplitudes ��/2: 0.16%, 0.4%and 0.7%; b) cyclic proportional limits, �prop and �yc, in dependence on plastic strain range. Thevalues of �prop are calculated in dependence on experimental values of dm; the values of �yc aredetermined with the aid of the �i-�p-hysteresis-loops and described by the dotted-line fit function.

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-tem-perature 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 thismodel, the plastic strain rate can be calculated as follows:

��p � CG exp �QG

RT

� �sinh

�VG �� i�RT

� �� 12�

The evolution equations of the internal parameters, �i and �VG (internal stress and acti-vation volume), are:

��i � HG E exp � �G�VG�i sign �� i�RT

� ��p �13�

and

� �VG � �K1�V2G��p� � K2�VG��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 mod-ulus is dependent on temperature (room temperature: E=116 GPa), the other constants,CG, QG, HG, �G, K1, K2, and the initial value of the activation volume �VG0, have tobe determined, e.g. by a parameter identification procedure. The original model isthree-dimensional, but here it is used only in the uniaxial case.

In cooperation with the Institut fur Allgemeine Mechanik und Festigkeitslehre(Prof. Steck) at the Technische Universitat Braunschweig, a set of parameters wasfound. This set of parameters takes the results of internal stress measurements and thedependence of the deformation behaviour on strain rate into account and is given by:

CG � 0�3670 10�5 s�1 � HG � 1�784 � �G � 0�3676 �

K1 � 47�20 MPa mol kJ�1 � K2 � 10�328 � and

�VG0 � 1�182 kJ mol�1 MPa�1 �

With these parameters, a good description of rapid hardening and cyclic saturation is pos-sible [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 thematerial in single-step tests. Furthermore, a relatively exact description of the hyster-esis-loop shape is possible. Some modifications seem to be necessary because the pa-rameters are valid only in a limited range of amplitudes and strain rates. More modifi-cations are needed to describe the stress-strain behaviour also in the case of incremen-tal-step tests and two-step tests with sufficient accuracy (for details: see Wittke [13]).


50

By Hatanaka and Ishimoto [28], another physically based model was proposed todescribe cyclic plasticity. In this model, assumptions are made concerning the evolutionof dislocation density and concerning the mean dislocation velocity. We have modifiedthe 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 copperand also for steady-state hysteresis-loops for 2.25Cr-1Mo [13]. An example for the lat-ter 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 ase.g. �J or �Jeff. According to the proposals of Dowling [32] and with the results ofShih and Hutchinson [33], it is possible to estimate �J in the case of various specimenand crack geometries. Schubert [18] measured the growth of cracks, which were ap-proximated as half circular surface cracks in circular specimens. Crack growth of2.25Cr-1Mo was measured with the ACPD method. In crack closure measurements, acrack closure parameter U was found, which is nearly constant: U=0.9 (compare Rieand Schubert [34] and Schubert [18]). Crack growth was successfully correlated with�J [18] and �Jeff [34], respectively.

The value U=0.9 was used also in the case of crack growth measurements ofedge cracks in flat specimens [13].

For the calculation of �J, characteristic values of the deformation behaviour areneeded. According to Dowling [32], the cyclic integral �J is calculated, e.g. in depen-dence on the cyclic hardening exponent n�. As proposed by Rie and Wittke [24], n� is


51

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, calculationwith the modified model of Hatanaka [28] (calculation: solid line; experiment: dotted line).

replaced by the exponent � of Equation (5). By this replacement, non-Masing behav-iour is taken into consideration. The values of � are calculated according to the methodproposed in [26] (for details: see Wittke [13]). With this method, the effective part of�J for edge cracks is estimated as:

�Jeff � 7�88U2 ��2

2E� 4�84��

�� p

1 � �

a � �15�

where a is the crack depth.The growth of edge cracks in flat specimens for the steel 2.25Cr-1Mo was mea-

sured with optical method. Tests with three different strain amplitudes (strain rate:�� 2 10�3 s�1) were performed. The relation between crack growth per cycle da/dNand �Jeff is described by:

da�dN � CJ��Jeff��J � �16�

where CJ and �J are constants. With an assumed initial crack depth and with a crackdepth, which defines failure, lifetime can easily predicted by integrating Equation (16)(compare Schubert [18]). With values for the characteristic parameters of hysteresis-loops, ��, ��p and �, the constants

CJ � 3�89 10�5 � �J � 1�16

were found (with da/dN in mm and �Jeff in Nmm/mm2). The correlation between da/dN and �Jeff is quite satisfactory as can be seen in Figure 3.10.


52

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

3.4 Creep-Fatigue Interaction

3.4.1 A physically based model for predicting LCF-lifeunder creep-fatigue interaction

In this section, the original model of the author proposed in 1985 [35] was described toillustrate in the following the modifications and the experimental verifications madesuccessively in the last years.

3.4.1.1 The original model

Unstable crack advance occurs if the crack progress per cycle, da/dN, becomes approxi-mately equal to the spacing of the nucleated intergranular cavities [35, 36]. The cracktip opening displacement �/2 may be seen as the upper bound to crack growth [36] andthe relation can be written as:

dadN

� �

2� �� 2r� � �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 ��el plus a contribution due to plastic deformation ��p and bythermally activated, time-dependent processes �c [37]:

� � a�K1��el � K2��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 cav-

ities on the number of cycles. In analogy to the Manson-Coffin relationship, we postu-late a constitutive equation for the cycle-dependent cavity nucleation under cyclic creepand low-cycle fatigue condition with superimposed hold time. Assuming that only theplastic strain imposed is responsible for cavity nucleation and disregarding stress de-pendence, the maximum number of cavities nmax is given by:

nmax � pN��p � �19�

where ��p is the plastic strain range, N is the number of cycles, p is the cavity nuclea-tion factor, and � is the cyclic cavity nucleation exponent. It was proposed that p wasidentical with the density of grain boundary precipitates. Since it was found that notevery precipitation necessarily produces a cavity, experimental constant � has beenused to adapt the observation in our first model. � was used as a fit factor to have bestresults in life prediction.


53

Cavity nucleation under creep-fatigue condition is favoured on grain boundariesperpendicular 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 cavitygrowth is controlled by diffusion. In the first model, the cavity growth model of Hulland 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 forevery cycle, and the result could be expressed in one compact equation for the life pre-diction. In this case, the lifetime is reached if:

a2�K1��el � K2��p � K3�c�

� �1��

pN��p� � 2

��4��gbDgb

��p��p

�� 2�kT

� t

0

��t� dt N�2�1

� ��

�� 21�

By integrating, it was assumed that the kinetics of the cavity growth in tension are thesame 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 onunzipping of cavitated material as the failure criterion. Therefore, in a first step of mod-ification, we use:

�

2� �� 2r � �22�

p was taken from direct experimental observation of cavities as will be shown in thefollowing. Therefore, it is not necessary to consider the influence of precipitation onthe nucleation of cavities, and nmax in Equation (19) could be replaced by the real den-sity of cavities on grain boundaries n.

The cavities nucleated by tensile stresses can be healed during periods of com-pressive stress if the compressive stress is applied for a long enough time. It has beenobserved that the time required to heal the cavity by compressive stress is up to sixtimes longer than the time to nucleate the cavity by tensile stress [39]. In a secondstep, the incomplete healing response has been modelled in dividing the rate of the ra-dius 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 wasintroduced. 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

were compared, and it could be shown that in case of the calculated lifetimes, the influenceof the model on the result is negligible [42]. The model of Riedel [41] is used in the fol-lowing 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 experimen-tally by means of stereometric metallography as will be shown in Section 3.4.2. Thesevalues have been used for the life prediction.

A furnace with window and special optics allows high magnification observationof the specimen surface continuously during the test with a video system and a subse-quent measurement of the crack growth and the crack tip opening. The value for cracktip opening was determined in a distance of 250 �m [43].

With this method, the fundamental assumption of the life prediction model aboutthe dependence of the crack tip opening displacement on the crack length and thestrain range expressed in Equation (18) could be experimentally checked. An exampleof the crack tip opening displacement in dependence on the crack length is shown inFigure 3.11. The slope of the straight line Cexp for the experiments ranges between0.043 and 0.058. The calculated slope determined with Equation (18) for the same ex-periment is Ccal =0.044. From that, it can be concluded that the calculation of the cracktip opening displacement in the original life prediction model leads to values, whichare in the right order of magnitude.

3.4.1.4 Life prediction

The fatigue life of high-temperature low-cyclic fatigue under arbitrary cyclic loadingsituations including wave shapes and hold time can be estimated using the unstablecrack advance criterion of the critical cavity configuration expressed in Equation (22).


55

Figure 3.11: Crack tip opening �250 �m vs. crack length.

The effective cavity spacing � can be estimated from Equations (19) and (20), thecrack tip opening displacement � from Equation (18). And the cavity radius r can beobtained by integrating the history-dependent cavity growth equation expressed inEquations (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 everycycle separately (Figure 3.12). From the radius of every nucleated and subsequentlygrown cavity, we calculate the mean value rm and compare �– 2rm with �/2 (Equation(22)) to get the critical life. Figure 3.13 shows the good agreement between experimen-tal data and predicted life using the pore growth model of Riedel [41].


56

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

Figure 3.13: Comparison of experimental life and predicted life.

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

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

3.4.2.1 Stereometric metallography

After low-cycle fatigue testing, the specimens were metallographically prepared forstereometric analysing for the density and cavity size distribution. For this purpose, thecavities were photographed by a Scanning Electron Microscope, and the cavity densityand 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 me-tallographic section are much different from the real cavity configuration in the volume.

For calculating the real cavity size distribution and density on the grain bound-aries, the following assumptions are made: All cavities are on boundaries oriented per-pendicular to the load axis with a maximum deviation of 30 �. All grains are of identi-cal size, which is the mean value (in this case 62 �m), and all cavities are sphericallyshaped.

The principal procedure is divided into two steps: First, the cavity size distribu-tion and the cavity density in the volume of the specimen were calculated. This wasdone from the corresponding values in the metallographic section by means of a nu-merical procedure. Spheres were placed in a given volume by means of the Monte Car-lo method. The spheres are randomly distributed. The size distribution of the sphereswas set as a logarithmic Gaussian distribution. The resulting size distribution was cal-culated in a section of the volume, which is designed as the imaginary metallographicsurface. The determined values of this section were compared with the experiment, andthis procedure was repeated by varying the density and the parameters of the Gaussiandistribution. This was done until the resulting density and the size distribution wereidentical to the values of the metallographic section.

Second, the real density on the grain boundaries ngb from the density in the vol-ume nv was calculated by means of a formula, which was provided by Needham andGladman [44]:

ngb � nvi

2q� �23�

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


57

3.4.2.2 Computer simulation

For the simulation, the cavities were placed one after another on a given area of thegrain boundary by the Monte Carlo method. After the nucleation of one additional cav-ity, the growth of all cavities on the grain boundary was calculated until the next wasformed. The growth of every cavity in our calculations depends on the spacing to the 6neighbouring pores in plane and is described by so extended diffusion-controlled cavitygrowth model. The extension is illustrated in Figure 3.14.

The advantage of the proposed model in comparison to the existing cavity growthmodels is the inhomogeneous distribution in plane. To calculate the cavity growth, itcan be assumed that the total vacancy flow to the cavity considered is the sum of theflows from all 6 segments as illustrated in Figure 3.14. We propose that the flow fromevery segment depends on the distance only to the nearest cavity within the segmentconsidered. This is analogous to existing 1-dimensional cavity growth models [45–48],but overestimates the vacancy flow because the contribution of the far distant cavitieswithin this segment is supposed to be the same as the nearest. The same considerationswill be applied for other segments. A fit factor is introduced, which takes this into con-sideration. 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, thecavity growth model proposed by Riedel [41] was chosen:

�r � 2��Db��b � �0�1 � ��k T h��q��r2

� �24�

�0 � 2�s

rsin � �25�

� 2r�

� �2

� �26�

q�� 2 ln � �3 � ��1 � � � �27�


58

Figure 3.14: Statistically distributed cavities in plane.

The meaning and the values of the constants for 304L are: h (�)=0.61 the relation be-tween the cavity volume and the volume of a sphere with radius r, �=1.21 ·10–29 m3

the atomic volume, �Db = 2 · 10–13 exp (–Q/RT)m3/s with Q=167 kJ/mol the grain bound-ary diffusion coefficient times the grain boundary width, 2�=70 � the void tip angle,�s =2 kJ/m2 the specific surface energy, k Boltzmann constant and T the temperature.

This differential equation (Equation (24)) was solved numerically. A possible coa-lescence has been taken into account. In this case, the two cavities were replaced by anew cavity, with the volume of both at the centre of the connecting line. As is shownbelow, this coalescence of cavities plays an important role in the case of fatigue be-cause the accumulated strain, which controls the cavity formation, is relatively highcompared to unidirectional tests. The cavity development during creep has also beensuccessfully 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 ofcycles 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 verifiedby the experiments as will be shown below.

Note that during the creep-fatigue, the stress is not constant, whereas it is con-stant in the case of pure creep. Therefore, in the calculation, the changing stress wastaken into consideration.

The cavities are formed continuously during the tension period of the cycle untilthe strain maximum is reached. During the hold period, stress relaxation occurs and nocavities are formed. The actual stress for calculating the cavity growth after the forma-tion of every single pore was taken directly from the experiment. During the compres-sion period, no further cavity formation occurs. Due to the negative stress, the cavitiesare shrinking. However, the influence of shrinkage is negligible for this kind of testwithout 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 isplotted versus the number of cycles. The cavity density during creep-fatigue dependson the number of cycles N by a power law as suggested before. With p=12 ·10–2 1/�m2 and �=0.4 in Equation (19), a good fit of the experimental data is possible (notplotted in the figure), and the fundamental idea about the cavity formation in the lifeprediction model is verified.

With this basic assumption about the cavity density in dependence on the numberof 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, largecavities 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 experimentand the simulation, which is the solid line, is given for different numbers of cycles. Inthe 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 inthe imaginary section by means of the method described in Section 3.4.2.1.


59


60

Figure 3.15: Experimentally detected cavity density and cavity density of the simulation vs. num-ber of cycles.

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

The proposed model and the constitutive equations for simulating the pore config-uration in the case of creep-fatigue leads to a good agreement between calculation andexperimentally detected cavity size distribution, and also the cavity density as shown inFigure 3.15. From Figure 3.15, one can draw the conclusion that the cavity coalescenceplays 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 re-sulting 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 dam-age development and predicting the fatigue life. But in the LCF-regime, failure is a lo-cal phenomenon, which takes place in front of the crack. Therefore, the strain has beenmeasured in front of the crack for giving the basis of a local application of materiallaws 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 grat-ing 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 amplitudeon the local strain distribution

The size of the highly deformed zone in front of the crack depends on the cracklength. This effect can be measured with this method. For both steels, the size of thehighly deformed zone increases with the crack length, which is shown in Figure 3.17by plotting R0.05 vs. crack length. The size of the highly deformed zone also dependson 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 totalstrain 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 propor-tional to the crack tip opening displacement �, Iino [12] observed the linear depen-dence of the highly deformed zone size R0.05 on the crack tip opening in the case oflow-cycle fatigue:

a � � � �a � � Shih and Hutchinson

R0�05 � � Iino � �28�

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


61

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 com-pression 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 andfor 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 oxi-dation of the crack surfaces. As a consequence, the crack surfaces can not return totheir original position of the previous cycle [51]. Therefore, the high deformationdeveloped at the tensile strain maximum cannot be completely reversed.


62

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

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

• A small shifting of the crack surfaces during opening of the crack may lead to anincomplete crack closure, and therefore to positive strain at the crack tip in com-pression.

• Due to the notch effect, stress and strain concentration occur at the crack tip duringcrack opening. However, when the crack closes, no stress concentration appearsand, as a consequence, the maximum stress at the crack tip in compression is equalto the total stress. Therefore, the stress at the crack tip in tension is higher than thestress 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 ofthe local strain measurement for crack growth calculation and life prediction. For thedemonstrated test, the crack advance is 4 �m per cycle. The size of the zone of positivemean strain in front of the crack is estimated at 1 mm. This means that the propagatingcrack advances for more than 250 cycles through a material, which has been cycled un-der 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 tensionhold 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 304Land different hold times. The same results are given for the ferritic steel in a paper ofthe authors [49].

The strain field depends on the hold time of the test, but remains the same duringthe hold period of each cycle within the accuracy of the measurement. In-situ monitor-ing of the crack advance and crack path indicates that the increase of crack growth rate


63

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

with tension hold time is related to a transition from a trans- to an intercrystalline crackpath. Metallographic observation of the microstructure shows that during tension, holdgrain boundary cavitation and microcrack formation occur.

We conclude that the microscopic changes of the material during the creep-fatiguesuch as the grain boundary damage lead to the change of the stress-strain behaviour infront of the crack. This phenomenon has to be emphasized particularly because the macro-scopic 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 isbased 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 Hasegawaand Ilschner [52] have detected a reduction of the strains in front of cracks in the caseof high temperature tension tests if cavities are formed.

3.5 Summary and Conclusions

The cyclic deformation behaviour at room temperature was investigated for copper andsteel 2.25Cr-1Mo. It can be concluded that for both materials, non-Masing behaviourhas to be taken into consideration. The investigation of the microstructure shows thatfor both copper and 2.25Cr-1Mo, dislocation cell structures were found for sufficienthigh strain amplitudes.

The deformation behaviour can be described by analytical relations. Especially forthe 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 de-fined and correlated with a microstructure-dependent proportional limit �prop. Calcula-tions with the physically based models of Steck and Hatanaka, respectively, show goodagreement with experimental results. The model of Hatanaka was modified by takingresults concerning the dislocation structure into account.

An application of test results in the field of fatigue fracture mechanics is shownby correlating da/dN and �Jeff.

The generalized life prediction model of the authors has the capability to predictlifetime of high temperature low-cycle fatigue under various wave shapes and holdtimes. Physically based constitutive equations for cavity nucleation and subsequentgrowth under variable loading histories are considered, and the unzipping of the cavi-tated grain boundary is taken as criterion for catastrophic failure. The crack tip openingdisplacement is seen as the upper bound to crack growth. These physically based as-sumptions in the model are verified by corresponding experiments.

The development of intergranular cavitation in austenitic steels can be simulated bythe 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 densityon grain boundaries are in accordance with the experiment. From this, it can be concludedthat the coalescence of neighbouring voids is very important for the cavity growth duringlow-cycle fatigue and is the main reason for the existence of relatively large cavities.


64

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

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[21] G. Hoffmann, O. Öttinger, H.-J. Christ: The Influence of Mechanical Prehistory on the Cyc-lic 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 Ap-plied Science, London, New York, 1992, pp. 106–111.

[22] C. E. Feltner, C. Laird: Cyclic Stress-Strain Response of F.C.C. Metals and Alloys – II: Dis-locations Structures and Mechanisms. Acta Metallurgica 15 (1967) 1633–1653.

[23] J. D. Morrow: Cyclic Plastic Strain Energy and Fatigue of Metals; Internal Friction,Damping, and Cyclic Plasticity. ASTM STP 378 (1965) 45–87.

[24] K.-T. Rie, H. Wittke: New approach for estimation of J and for measurement of crackgrowth at elevated temperature. (To be published in: Fatigue Fract. Mater. Struct. Vol. 19(1996).)

[25] U. Peil, J. Scheer, H.-J. Scheibe, M. Reininghaus, D. Kuck, S. Dannemeyer: On the Behav-iour of Mild Steel Fe 510 under Complex Cyclic Loading. This book (Chapter 10).

[26] H. Wittke, J. Olfe, K.-T. Rie: Description of Stress-Strain Hysteresis Loops with a SimpleApproach. (To be published in: Int. J. Fatigue (1996/97).)

[27] J.C. Tsou, D. J. Quesnel: Internal Stress Measurements during the Saturation Fatigue ofPolycrystalline Aluminium. Mat. Sci. and Engin. 56 (1982) 289–299.

[28] K. Hatanaka, Y. Ishimoto: A Numerical Analysis of Cyclic Stress-Strain-Response in Termsof Dislocation Motion in Copper: In: H. Fujiwara, T. Abe, K. Tanaka (Eds.): ResidualStresses – III, Elsevier Applied Science, 1991, pp. 549–554.

[29] J. Polak, M. Klesnil, J. Helesic: The Hysteresis Loop: 2. An Analysis of the Loop Shape.Fatigue of Engineering Materials and Structures 5(1) (1982) 33–44.

[30] H. Schlums, E.A. Steck: Description of Cyclic Deformation Process with a Stochastic Mod-el for Inelastic Behaviour of Metals. Int. J. of Plasticity 8 (1992) 147–159.

[31] R. Gerdes: Ein stochastisches Werkstoffmodell fur das inelastische Materialverhalten metal-lischer Werkstoffe im Hoch- und Tieftemperaturbereich. Braunschweiger Schriften zur Me-chanik 20 (1995).

[32] N.E. Dowling: Crack Growth During Low-Cycle Fatigue of Smooth Axial Specimens; Cyc-lic Stress-Strain and Plastic Deformation Aspects of Fatigue Crack Growth. ASTM STP637 (1977) 97–121.

[33] C. F. Shih, J.W. Hutchinson: Fully Plastic Solutions and Large Scale Yielding Estimates forPlane Stress Crack Problems. Journal of Engin. Mat. and Technol., Oct. 1976, Transactionsof ASME, pp. 289–295.

[34] 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, ElsevierApplied Science, pp. 575–580.

[35] K.-T. Rie, R.-M. Schmidt, B. Ilschner, S.W. Nam: A Model for Predicting Low-Cycle Fa-tigue Life under Creep-Fatigue Interaction. In: H.D. Solomon, G.R. Halford, L.R. Kai-


66

sand, B. N. Leis (Eds.): Low Cycle Fatigue, ASTM STP 942, American Society for Testingand Materials, Philadelphia, 1988, pp. 313–328.

[36] G. J. Lloyd: High Temperature Fatigue and Creep Fatigue Crack Propagations Mechanics,Mechanisms and Observed Behaviour in Structural Materials. In: R. P. Skelton (Ed.): Fa-tigue at High Temperatures, Applied Science Publishers, London, New York, 1983, pp.187–258.

[37] R.-M. Schmidt: Lebensdauer bei Kriechermudung im Low-Cycle Fatigue Bereich. Disserta-tion TU Braunschweig, VDI Fortschritt-Berichte Nr. 47 (1988).

[38] D. Hull, D.E. Rimmer: The Growth of Grain-Boundary Voids Under Stress. PhilosophicalMagazine 4 (1959) 673–687.

[39] B. K. Min, R. Raj: Hold Time Effects in High Temperature Fatigue. Acta Metall. 26 (1978)1007–1022.

[40] H.E. Evans: Mechanisms of Creep Fracture. Elsevier Applied Science Pub. LTD., 1984,pp. 251–263.

[41] H. Riedel: Fracture at High Temperatures. Springer-Verlag, Berlin Heidelberg, 1987.[42] K.-T. Rie, J. Olfe: A physically based model for predicting LCF life under creep fatigue in-

teraction. In: K.-T. Rie (Ed.): Proc. 3rd Int. Conf. on Low Cycle Fatigue and Elasto-PlasticBehaviour of Materials, Elsevier Applied Science, London, New York, 1992, pp. 222–228.

[43] K. Tanaka, T. Hoshide, N. Sakai: Mechanics of Fatigue Crack-Tip plastic blunting. Engi-neering Fracture Mechanics 19 (1984) 805–825.

[44] N.G. Needham, T. Gladman: Nucleation and growth of creep cavities in a Type 347 steel.Mat. science 14 (1980) 64–66.

[45] S. J. Fariborz: The effect of nonperiodic void spacing upon intergranular creep cavitation.Acta metall. 33 (1985) 1–9.

[46] S. J. Fariborz: Intergranular creep cavitation with time-discrete stochastic nucleation. Actametall. 34 (1986) 1433–1441.

[47] J. Yu, J.O. Chung: Creep rupture by diffusive growth of randomly distributed cavities – I.Instantaneous cavity nucleation. Acta metall. 38 (1990) 1423–1434.

[48] J. Yu, J.O. Chung: Creep rupture by diffusive growth of randomly distributed cavities – II.Continual cavity nucleation. Acta metall. 38 (1990) 1435–1443.

[49] 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 Measure-ments in Non-Uniform Fields at Elevated Temperatures, March 14–15, Berlin, 1996.

[50] K.-H. Schwalbe: Bruchmechanik metallischer Werkstoffe. Carl Hanser Verlag, Munchen,Wien, 1980.

[51] T. Ericsson: Review of oxidation effects on cyclic life at elevated temperature. Canadian me-tallurgical quarterly 18 (1979) 177–195.

[52] T. Hasegawa, B. Ilschner: Characteristics of crack tip deformation during high temperaturestraining of austenitic steels. Acta metall. 33(6) (1985) 1151–1159.

References

67

4 Development and Application of Constitutive Modelsfor the Plasticity of Metals

Elmar Steck, Frank Thielecke and Malte Lewerenz*

Abstract

The macroscopic behaviour of crystalline materials under mechanical or thermal load-ings is determined by processes in the microregion of the material. By a combinationof models on the basis of molecular dynamics and cellular automata, it seems possibleto simulate numerically the formation of internal structures during the deformation pro-cesses. The stochastical character of these mechanisms can be considered by modellingthem as stochastic processes, which result in Markov chains. By a mean value formula-tion, this leads to a macroscopic model consisting of non-linear ordinary differentialequations. The determination of the unknown material parameters is based on a Maxi-mum-Likelihood output-error method comparing experimental data to the numerical si-mulations. With Finite-Element methods, it is possible to use the material models forthe design of components and structures in all fields of technical application and forthe numerical simulation of their behaviour under complex loading situations.

4.1 Introduction

Metallic materials show, like other crystalline substances, typical macroscopic re-sponses on mechanical loading, which are caused by processes on the microscale. Fig-ure 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 berecognized by the fact that plastic flow occurs after load reversal at significantly lowerstresses than those, from which the load reversal was done. For the technical use ofmetallic materials, the description of this kind of processes in material models is ofhigh importance.

68

* Technische Universitat Braunschweig, Institut fur Allgemeine Mechanik und Festigkeitslehre,Gaußstraße 14, D-38106 Braunschweig, Germany



The moving of dislocations is the main microscopic mechanism responsible forthe plastic deformations in metallic materials. In the following, a stochastic model ispresented, which is able to consider hardening and recovery processes by means ofMarkov chains.

During the deformation process, the dislocations arrange in a hierarchy of struc-tures such as walls, adders or cells. This forming of structures influences the macro-scopic behaviour of the materials considerably. The principle of cellular automata incombination with the method of molecular dynamics is used for the numerical simula-tion of these processes.

For the material parameter identification, the minimization of the Maximum-Like-lihood costfunction by hybrid optimization methods parallelized with PVM is consid-ered. With a multiple shooting method, additional information about the states can betaken into account, and thus the influence of bad initial parameters will be reduced. Forthe analysis of structures like a notched flat bar, the Finite-Element Program ABAQUSis used in combination with the user material subroutine UMAT. The results are com-pared with experimental data from grating methods.

4.2 Mechanisms on the Microscale

The movement of dislocations and the connected plastic deformations caused by exter-nal loading is determined by two important activation mechanisms. The stress activa-tion is caused by the external loads. The thermal activation supports at elevated tem-peratures the dislocation movements and therefore the plastic deformations.

4.2 Mechanisms on the Microscale

69

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

Figure 4.2 shows schematically the obstacles, which resist the dislocation move-ments on the microscale in the form of barrier potentials U*, the possible position – de-termined by temperature – of the dislocations relative to these barrier potentials, andthe effect of an external load and a temperature increase on the energetic situation ofthe dislocations. It is visible that the potential U � of the external forces by superposi-tion changes the potentials of the actual obstacles so that the dislocation movement inthe direction of the applied stress is more probable than in the opposite direction, andthat the thermal activation supports this process [1–4].

The barriers, which oppose the dislocation movements, are on the one side givenby the crystalline structure of the material itself, on the other hand, foreign atoms andgrain boundaries can form obstacles. One of the most important reasons for the hinder-ing of the dislocations, however, are the dislocations themselves. During plastic defor-mation, continuously new dislocations are produced. In the beginning, the ability of thematerial for deforming plastically is increased. With increasing dislocation density, amutual influence of the lattice disturbances occurs, which results in isotropic hardening.

Due to the lattice distortions connected with the plastic deformation, elastic en-ergy 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 stres-ses, however, support the dislocation movements in the opposite direction and result ine.g. the Bauschinger effect. At elevated temperatures – above half of the melting tem-perature of the material –, thermally activated reorganization processes in the crystalsoccur, which reduce the mutual hindering of the dislocations and result macroscopicallyin recovery.

Significant magnitudes for these processes are given in Figure 4.3, which shows adislocation, which is influenced by other dislocations. The shaded area is a measure forthe activation volume �V � bA, which decreases in size with increasing isotropic hard-ening. The Burgers vector b determines with his orientation relative to the dislocationline the character of the dislocation. �w is the density of the so-called forest disloca-tions, 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 impor-tant dislocation mechanisms for different regions of the homologous temperature T�Tm.

4 Development and Application of Constitutive Models for the Plasticity of Metals

70

Figure 4.2: Stress and thermal activation of dislocation motion.

4.3 Simulation of the Development of Dislocation Structures

For unidirectional as well as for cyclic plastic deformation, it is observed that disloca-tion structures are developed in the shape of e.g. adders or dislocation cells, which in atypical 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 macro-scopic behaviour of the materials considerably, the simulation of these self-organizationprocesses can result in valuable information for the choice of formulations for the mod-elling of processes on the microscale.

The interaction of a large number of identical particles is the basic idea for thedefinition of cellular automata. It is an idealization of real physical systems, wherespace as well as time are discrete. A cellular automaton is completely characterized bythe following four properties [5]: geometry of the cell arrangement, definition of aneighbourhood, definition of the possible states of a cell, and evolution rules. Each cellcan during the evolution in time only assume values (states) out of a finite set. For all

4.3 Simulation of the Development of Dislocation Structures

71

Figure 4.3: Activation volume and forest dislocations.

Table 4.1: Activation volume depending on deformation mechanism and temperature.

Mechanism Temperature Activation volume

Climbing >0.5 b3 remains constant during deformation

Movement ofdislocation jumps

>0.5 10–1000 b3, the value of the activation volume decreasesduring deformation

Cross slip 0.2–0.4 10–100 b3, the value remains approximately constantduring deformation

Cutting ofdislocations

>0.3 1000 b3, the activation volume decreases due to increaseof the density of forest dislocations with increasingdeformation

cells, the same evolution rules are valid. The change in state of a cell depends on itsown state and those of the neighbouring cells.

Opposite to the usual assumptions for cellular automata, where the state of a cellonly depends on the states of the next neighbours, for the simulation of dislocationmovements, it has to be taken into account that the dislocations possess long-range act-ing stress fields. With this model, it is possible to compute the dynamics of some thou-sand 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 cho-sen, assumes a grid of rectangular cells, which can be occupied by edge- or screw-dis-locations with positive or negative sign [5]. The transition rules are: A positive or nega-tive occupied cell becomes an empty cell if the dislocation in the cell will move due tothe acting forces to a neighbouring cell or if an annihilation with a dislocation in aneighbouring cell occurs. The step width of a dislocation is always one cell size pertime 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 theforces acting on a dislocation, a larger neighbourhood is necessary due to the long-range acting stress of the dislocations. The balance of forces decides, if and in whichdirection a dislocation will move. It is computed for each time step and each disloca-tion for both degrees of freedom.

A much more realistic simulation for the development of dislocation structures isobtained from models, which consider several glide planes [6]. Figure 4.5 shows a two-dimensional projection for the glide system for a cubic face-centred lattice, and model-ling of the glide processes on this system with three glide directions under angles of re-spectively 60�.

The simulation results in wall- and labyrinth-structures of the dislocations (Fig-ure 4.6). An extension of the model with consideration of vacancies and a suitable velo-city law is under progress.


72

Figure 4.4: Characteristic dislocation structures.

4.4 Stochastic Constitutive Model

The description of the processes responsible for plastic deformations shows that theyare strongly stochastic. Figure 4.7 shows for a simplified case for processes at hightemperatures, under consideration of kinematic hardening only, the used stochasticmodel [1, 2].

Over the state axis, which represents the value of the kinematic hardening �kin,and therefore the strength of the obstacles resisting the dislocation movements, the dis-tribution of the “flow units” (dislocations, dislocation packages or grain boundaries) isgiven. The effect of the external stress is reduced by the hardening stress, thereforeonly the effective stress �eff � �� kin is responsible for the dislocation movements.Depending on �eff, a hardening probability


73

Figure 4.5: Cell arrangement and neighbourhood of simulation model.

Figure 4.6: Simulation of dislocation structures.

V � c1 exp � ��V�kin

RTsign�eff

� ��ie �1�

is formulated. This transition probability is based on the condition that thermal activa-tion of the dislocations can be taken as an empirical Arrhenius function. R is the gasconstant and c1� �� V are constants, which have to be determined by experiment. Itcan be seen that the transition probability from a certain hardening state to the nexthigher decreases with increasing hardening.

Hardening is opposed by a recovering process according to:

E � c2 exp � F0

RT

� � ��kin��0

� �m

sinh�V�kin

RT

� �� 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 distortionsincreases with increasing hardening. It supports the recovery process. Therefore, thetransition probabilities for recovery increase with increasing hardening.

The model simulates hardening and recovery by transitions of dislocations at abarrier strength �kin�i to higher barriers �kin�i�1 and lower barriers �kin�i�1� The probabil-ity that a flow unit remains in the actual position is given by:

B � 1 � V � E � �3�

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


74

Figure 4.7: Stochastic model for high temperatures.

S �

1 � V1 E2

V1 B2� ��

0

V2� ��

Ei

� ��

Bi� ��

Vi� ��

Ek�1

0 � ��

Bk�1 Ek

Vk�1 1 � Ek

��

��

� �4�

The change of the structure, which is described by the state vector z, during onetime step �t is given by the Markov chain:

z�t � �t� � S z�t� � �5�

For constant stress and temperature (homogeneous process), the state vector aftern time steps is given by z�t0 � n�t� � Snz�t0�� The stochastic matrix given by Equation(4) can be transformed to principal axes and yields then:

�S � M�1SM �1 0 0 00 �2 0 0

0 0 � ��

00 0 0 �n

��

�� 6�

where M is the modal matrix, i.e. the matrix of the columnwise arranged eigenvectorsof the matrix S. Due to the fact that the maximal principal value of stochastic matricesis 1 and all other eigenvalues have magnitudes <1, it is visible that their magnitudesdecrease with increasing time, and the eigenvalue connected with the maximal eigenva-lue 1 represents a stationary state. The other principal values are responsible for transi-ent processes [1–3].

An extension of the stochastic model, which allows for the simultaneous consid-eration of the development of activation volume �V and kinematic stress �kin is givenin Figure 4.8. Thus, isotropic and kinematic hardening spread a state plane, which al-lows that with the distribution of the flow units, the state determined by both hardeningtypes can be considered. The transition probabilities for the description of the develop-ment of the isotropic and kinematic hardening consider mutually the influences givenby the other hardening process [3, 4].

By a mean value formulation, the stochastic model is transformed in a macro-socpic continuum mechanical material model, which takes a form similar to other mod-els given in literature. This approach leads to a non-linear system of ordinary differen-tial equations for the inelastic strain �, the kinematic back stress �kin and the activationvolume �V:


75

��ie � C exp � F0

RT

� � ��eff��0

� �n

sinh�V�eff

RT

� �� 7�

��kin � H � exp � ��V�kin

RTsign�eff

� ��ie�R� exp �F0

RT

� � ��kin��0

� �m

sinh�V�kin

RT

� ��

�8�

� �V � �K1�V2��ie� � K2�V ��ie� � �9�

The material behaviour is described by a relation for the inelastic strain rate, where theactual values for isotropic and kinematic hardening occur as internal variables. Thisgeneral form of the constitutive equations is also the basis for the development of ahierarchical model classification [7]. A concrete model must be chosen with respect tothe intended application purpose. The values C� n� H� �� R�� m� K1� K2 and �V0 arematerial parameters, which have to be determined by comparison with experimental re-sults. The parameter identification, which consists in integrating the non-linear, ordi-nary differential equations for varying parameter sets and by appropriate optimizationmethods to search for the optimal parameter sets, deserves special recognition in aspectof the used mathematical methods [7, 8]. An additional scaling of the functions like

exp �F0

R1T� 1T0

� �is necessary to improve the parameter identifiability and the

macroscopical interpretations.


76

Figure 4.8: Distribution function for �kin and �V=bA.

4.5 Material-Parameter Identification

4.5.1 Characteristics of the inverse problem

Under the assumption of normal distributed measurement errors with zero mean andknown measurement-covariance matrix C�ti�� the costfunction is:

L2�x� p� � 12��r��2 � 1

2

�ni�1

�z�ti� � x�ti�TC�1�z�ti� � x�ti��min � �10�

The minimization of this weighted least squares function yields a Maximum-Likelihoodestimate of the parameters, which reproduces the observed behaviour z of the real pro-cess with maximum probability [9].

Typical features of the identification are that the constitutive model is not onlyhighly non-linear in states x, but also in parameters p. Due to incomplete measurementinformation, the problem is ill-conditioned, parameters are highly correlated. Becauseof unbalanced parameters, the model may change its characteristics and becomes stiffor even pathological. Since replicated tests for the same laboratory conditions show a sig-nificant scattering and thus incompatibility of the data, this uncertainty must be taken intoaccount 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 im-portant informations about the deformation process and thus can be used to get morereliable parameters. In general, there are no (complete) measurements for the internalstates. However, engineers have a lot of additional apriori-information, which should beused to improve the model prediction capacity. Although it is possible to formulate ad-ditional weighted least-squares terms for the Maximum-Likelihood function, a muchmore 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 gridand to treat the discretized model equations as non-linear constraints of the optimiza-tion problem. The initial state estimates at the nodes of the grid allow to make efficientuse 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

Adaptive Evolution Strategies attempt to imitate the organic evolution process,e.g. a collective learning of a population with the principles mutation, recombinationand selection [16]. The self-learning process of strategy parameters adapts this optimi-zation procedure to the local topological requirements. Since it is possible to overcomelocal minima with a special destabilization method, evolution strategies even work withbad initial model parameters [8].

Damped Gauß-Newton methods are most widely used for the minimization ofnon-linear least-squares functions [7, 11]. Starting from initial parameters p0, improvedparameters are iteratively obtained by the solution of a linear least-squares problem lin-earized about pk. The steplength parameter �k is chosen to enforce the convergenceproperties:

12�� r�pk� � Jk�pk��2 min with pk�1 � pk � �k� pk � �11�

solution with pseudo inverse: � pk � �J��pk�r�pk� �

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 consumingpart of the identification. Since the calculation is very closely related to the numericalintegration of the differential equations and the available accuracy, the sensitivity analy-sis may be a critical point. Three different concepts are used to generate the sensitivitymatrix. The commonly used finite difference approximation:

�xi�pj

� xi�pj � �pj� � xi�pj��pj

�12�

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

�x�p

�� f

�x

�x�p

� �f

�p

� �13�

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


78

Figure 4.9: Principle of multiple shooting.

(IND) [11]. This means that the internally generated discretization scheme of the inte-grator 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 reliabilityis 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 experimen-tal designs. Based on design criteria like the minimization of det�JTJ��1� differentmethods have been considered and tested for typical growth function and a fundamen-tal constitutive model. These studies also show that the bad identifiability of the in-verse 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 pa-rameter identification problem. In addition, evolution strategies and multiple shootingprovide inherent parallelism on a high level. Thus, efficient parallel computation ofmodel functions and derivatives can be easily performed on a workstation-cluster withPVM (Figure 4.10).


79

Figure 4.10: Parallel simulation concept.

Since the development and identification of anisotropic damage models becamemore and more important, a three-dimensional Finite-Element program was coupledwith the estimation software by a special interface. The flexibility and modular struc-ture of this approach may be very useful for a lot of other applications, e.g. structureoptimization.

For the application of the damped Gauß-Newton method, the Internal NumericalDifferentiation was adapted to the Finite-Element analysis. Thus, not only the simula-tion results but also the sensitivities have to be transferred. The results of the simula-tions are compared with experimental strain fields obtained by grating methods [17].


80

Figure 4.11: Creep tests for aluminium.

4.5.6 Comparison of experiments and simulations

A lot of different materials like pure aluminium, pure copper or the austenitic steelsAISI 304 and AISI 316 have been extensively studied.

Figure 4.11 shows some results of parameter identifications for aluminiumAl 99.999. The temperature regime was between 500 �C and 700 �C. Since only mono-tonic tests were evaluated, a constitutive model with only one structure variable for theinternal stress is used. The parameters were identified for the given stresses simulta-neously 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 ofseven strain-controlled cyclic tests at room temperature [18]. Two strain rates ��=10–4,10–3, and a multitude of strain amplitudes ��2 =0.2–0.7% are examined.

For this application of the stochastic constitutive model, the special characteristicsof the material and the measurements have to be considered. In the low-temperature re-gime, hardening is the most important phenomenon, while the recovery influence isnegligible. In contrast to high temperatures, metal physical results also indicate that the


81

Figure 4.12: Cyclic tests for copper.

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 usuallyconsist of only one observed trajectory for each temperature and loading condition.Nevertheless, replicated tests for the same laboratory conditions show a significant scat-tering and thus incompatibility of the measurements (Figure 4.13). Based on a statisti-cal analysis, this uncertainty can be taken into account for more reliable modelling andparameter identification.

The modelling of the experimental uncertainties is based on the scattering of theparameters or the initial values (Figure 4.14) [10].

Based on these concepts, realistic simulations of the uncertainties in experimentaldata due to measurement errors and scattering are possible [7].


82

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

4.6 Finite-Element Simulation

The aim of using constitutive models is to predict the behaviour of metallic structuresunder mechanical and thermal loading. This requires the solution of a coupled initial-boundary value problem, given by the momentum equilibrium and the constitutiveequations. Since the boundary value problem is usually solved by the Finite-ElementMethod (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 im-plementation are discussed for application of the user subroutine UMAT.

The developed method of implementation is described in Section 4.6.1. The maincharacteristics of this method are its applicability to any unified constitutive model ofthe class described above and to small as well as to large deformations theory. In Sec-tion 4.6.2, some numerical and experimental results are given, which show that themodel 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 sys-tem of non-linear ordinary differential equations (CSNODE), which builds an initialvalue problem. Its solution to a time increment can be embedded in an incremental Fi-nite-Element formulation with displacement approach, leading to the well known impli-


83

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

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, thecomputational cost of the FEM-simulation can be minimized by optimizing the numeri-cal 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 sub-sections (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 takesthe form:

��ij � fij��kl� �kl� �kinkl ��V� � �14�

��kinij � fij��kl� �kin

kl ��V� � �15�

��V � f ��kl� �kinkl ��V� � �16�

where �ij is the Cauchy stress tensor, �kinij is the back stress tensor and ��ij is the defor-

mation rate tensor. Each of these symmetric tensors is defined by six independent com-ponents, so that the whole CSNODE contains thirteen scalar equations.

Since the v. Mises equivalent stress

�v ��32� ij�

ij

�17�

just depends on the deviatoric stresses, the inelastic part of the tensor equations arealso purely deviatoric. Therefore, the deviatoric rates ��ij and ��kin

ij can be described insome interval �t0� t0 � �t as a linear combination of the three deviatoric tensors�ij�t0�� kin

ij �t0� and ��ij�t0�, as long as ��ij is constant in �t. Using a suitable transforma-tion, the deviatoric rates can be expressed in a subspace by only three independentcomponents. After this transformation, the initial value problem (Equations (14) to(16)) can be written as:

�yi � fi�yj� � yi�t0� � yi�0 � i� j � 1 � � � n � �18�


84

and contains only seven scalar equations. Hornberger [21] shows that the subspace di-mension 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 uniax-ial states as well. Although the number of scalar equations cannot be reduced in thesecases, the main advantage is that the transformed model equations in the subspace areof identical form for each of these cases. Based on this fact, the model implementationfor 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 theoryas 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 beintegrated numerically. In oder to choose an appropriate integration algorithm, the inte-gration 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 integrationorder p, which describes the discretization error R in dependency of the step size h byR � hp (for an overview see [23, 24]). There are:

• methods with fixed integration order like multi-step methods, Runge-Kutta meth-ods, and Taylor series methods, and

• methods with variable integration order like extrapolation methods.

Extrapolation methods are efficient only for high integration tolerances, while multi-stepmethods loose efficiency for small integration intervals. The use of Taylor series methodsis not practicable since it requires higher derivatives of the CSNODE, which are usuallynot given directly. So, explicit and implicit, Runge-Kutta methods are widely used for theintegration of constitutive equations in FEM-analysis (see e.g. [19, 21, 25]).

Butcher [26] pointed out that the mentioned methods with fixed integration ordercan be combined to get new classes of integration methods. For example, so-called Ro-senbrock methods result from the combination of Runge-Kutta methods and Taylor se-ries methods based on the first derivative of the CSNODE (also called the Jacobean ofthe system). The main advantage of these methods is their unconditional stability – asin implicit Runge-Kutta methods – that is reached with an explicit algorithm withoutany iteration process. Rosenbrock methods as well as Runge-Kutta methods can be de-signed as embedded integration formulae, which lead directly to a method of internalerror estimation without additional numerical cost.


85

Verner [27] proposed families of embedded explicit Runge-Kutta processes, whichallow to rise integration order p without dismissing the results of an integration with alower starting order p0. If this concept is used with Rosenbrock methods, the resultingintegration process is kind of an optimal numerical integration method for constitutiverate 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 inter-

val 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

In non-linear implicit FEM-analysis, the tangent modulus��ij��kl

is used to compute the

element stiffness matrix, which is the tangent operator for the applied Newton iterationmethod. Due to the necessity of numerical integration, the stress increment ��ij is adiscrete value and so, the partial derivative cannot be built analytically. Therefore, ithas to be approximated numerically too. This can be done by an Internal NumericalDifferentiation (IND), which was proposed by Bock [11]. Illustratively, IND means tocompute the derivative of the numerical integration algorithm, which leads to the dis-crete stress increment. The IND computes an approximation of the partial derivativethat is of similar relative exactness as the solution of the integration itself.

4.6.2 Deformation behaviour of a notched specimen

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


86


87

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

4.7 Conclusions

The mechanisms on the microscale of crystalline materials can be examined on differ-ent scales of magnitude. Starting from a scale, where the processes are described byhelp of activation energies and activation volumes as mechanically and thermally acti-vated, it is possible to consider their stochastical nature by stochastic processes, fromwhich by mean value considerations, a transition to macroscopic material equations ispossible.

To support the formulation of these models, simulations can be useful, which con-sider the multi-particle properties of the processes, and use the methods of cellularautomata or molecular dynamics.

For the numerical simulation and the parameter identification, a variety of sophis-ticated methods have been considered. The results show that it is possible to use thematerial 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 Pro-cesses. Res. Mechanica 25 (1990) 1–19.

[3] H. Schlums: Ein stochastisches Werkstoffmodell zur Beschreibung von Kriechen und zyk-lischem Verhalten metallischer Werkstoffe. Dissertation TU Braunschweig, BraunschweigerSchriften zur Mechanik 5 (1992).

[4] R. Gerdes: Ein stochastisches Werkstoffmodell fur das inelastische Materialverhalten metal-lischer 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 zellularen Automaten. Dissertation TU Braunschweig, Braun-schweiger Schriften zur Mechanik 4 (1992).

[6] D. Sangi: Versetzungssimulation in Metallen. Dissertation TU Braunschweig, 1996.[7] F. Thielecke: Parameteridentifizierung von Simulationsmodellen fur das viskoplastische Ver-

halten 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 fur metallische Bau-

teile 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 Scatter-ing of Experimental Data. ZAMM Z. angew. Math. Mech. 76 (1996).

[11] H.G. Bock: Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtli-nearer Differentialgleichungen. Bonner Mathematische Schriften, Bonn, Vol. 183 (1985).

[12] J. Schloder: Numerische Methoden zur Behandlung hochdimensionaler Aufgaben der Para-meteridentifizierung. Dissertation Universitat Bonn, 1987.


88

[13] F. Thielecke: Ein Mehrzielansatz zur Parameteridentifizierung von viskoplastischen Werk-stoffmodellen. ZAMM Z. angew. Math. Mech. 76 (1996).

[14] R. Jategaonkar, F. Thielecke: Evaluation of Parameter Estimation Methods for UnstableAircraft. AIAA Journal of Aircraft 31(3) (1994).

[15] R. Gerdes, F. Thielecke: Micromechanical development and identification of a stochasticconstitutive model. ZAMM Z. angew. Math. Mech. (1996).

[16] I. Rechenberg: Evolutionsstrategie ’94, Werkstatt Bionik und Evolutionstechnik, Band 1.Frommann-Holzboog, Stuttgart, 1994.


[18] K.-T. Rie, H. Wittke: Inelastisches Stoffgesetz und zyklisches Werkstoffverhalten im LCF-Be-reich, Arbeits- und Ergebnisbericht 1991–1993. Subproject B4, Collaborative ResearchCentre (SFB 319), 1993.

[19] E. Hinton, D.R. J. Owen: Finite Elements in Plasticity: Theory and Practice. PineridgePress, Swansea, 1980.

[20] M. Lewerenz: Zur numerischen Behandlung von Werkstoffmodellen fur zeitabhangig plas-tisches Materialverhalten. Dissertation TU Braunschweig, 1996.

[21] K. Hornberger: Anwendung viskoplastischer Stoffgesetze in Finite-Element-Programmen.Dissertation Universitat 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 (NonstiffProblems). 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 Stress-Strain Relations. Int. J. Numer. Meth. Eng. 24 (1987) 893–911.

[26] J.C. Butcher: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta andGeneral 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.

References

89

5 On the Physical Parameters Governingthe Flow Stress of Solid Solutionsin a Wide Range of Temperatures

Christoph Schwink and Ansgar Nortmann*

Abstract

At sufficiently low temperatures, host and solute atoms remain on their lattice sites. Thecritical flow stress �0 is governed by a thermally activated dislocation glide (Arrheniusequation), which depends on an average activation enthalpy, �G0, and an effective obsta-cle concentration, cb. The total flow stress � is composed of �0 and a hardening stress �d,which increases with the dislocation density �w in the cell walls according to �d � ��w�1�2.At higher temperatures, the solutes become mobile in the lattice and cause an additionalanchoring of the glide dislocations. This is described by an additional enthalpy �g in theArrhenius equation. In the main, �g depends on the activation energy Ea of the diffusingsolutes and the waiting time tw of the glide dislocations arrested at obstacles. Three dif-ferent diffusion processes were found for the two f.c.c.-model systems investigated,CuMn and CuAl, respectively. Under certain conditions, the solute diffusion causes in-stabilities in the flow stress, the well-known jerky flow phenomena (Portevin-Le Chateliereffect). Finally, above around 800 K in copper based alloys, the solutes become freelymobile and �0 as well as �g vanish. In any temperature region, only a small total numberof 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 � in metallic systems of model character over agiven wide range of temperatures and strain rates.

90

* Technische Universitat Braunschweig, Institut fur Metallphysik und Nukleare Festkorperphysik,Mendelssohnstraße 3, D-38106 Braunschweig, Germany



Any theory describing plastic deformation modes of such systems will have tomake use of these – and only these – parameters. As model systems we choose singlephase binary f.c.c. solid solutions. They are on the one hand simple, macroscopicallyhomogeneous materials, on the other hand exhibit all basic processes, which occur alsoin more complex alloys of technical interest. To cover a wide range of different charac-teristics 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 withsolute concentration and the tendency for short range ordering. The systems have incommon a metallurgical simplicity and a large range of solubility. The samples usedwere rods of polycrystals, for CuMn also of single crystals oriented either for single orfor multiple ([100], [111]) glide.

For low enough temperatures, i.e. roughly below room temperature, host and soluteatoms remain on their lattice sites in our systems. Then, the flow stress is recognized toconsist of two additive parts, which are in single crystals the critical resolved shear stress�0, and the shear stress �d produced by strain hardening. �0 is best examined on crystalsoriented for single glide, while results on �d originated from studies on [100] and [111]crystals. The parameters governing �0 and �d are discussed in Section 5.2.

At higher temperatures, the solute atoms become increasingly mobile and start todiffuse to sinks, e.g. dislocations. As a consequence, an additional anchoring of glidedislocations occurs, known as dynamic strain ageing (DSA), which results in an addi-tional contribution to flow stress, ��DSA, and in a decrease of the strain rate sensitivity(SRS) with increasing deformation. If the SRS reaches a critical negative value, jerkyflow sets in, the so-called Portevin-Le Chatelier (PLC) effect. The mechanisms induc-ing DSA and the relevant parameters represent the main part of project A8 and are re-ported 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.

5.1 Introduction

91

Table 5.1: Metallurgical and physical properties of ��Al and ��Mn.

��Al ��Mn

Misfit ��

� �0.067(weak hardening)

� �0.11(strong hardening)

Bulk diffusion �� eV��

�� eV��

��

Stacking fault energy strongly decreasing withincreasing ��

independent of ��

Slip character 5 � � � 10 at% Al planar 0.5 � � � � at% Mn homogeneous

Short range order marked and increasing with �� negligible up to � at%

5.2 Solid Solution Strengthening

In the frame of the project, invited overviews on “Hardening mechanisms in metals withforeign 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, �0

A detailed investigation on CuMn [4] showed that the thermally activated process gov-erning �0 is phenomenologically completely characterized by two parameters, the aver-age activation enthalpy, �G0, of the effective glide barriers, and the concentration ofthe latter, cb. This concentration resulted about 20 times smaller than the solute concen-tration, cMn. For �G0, values around 1.3 eV were found.

The magnitude of cb and �G0 suggest the effective glide barriers to consist ofcomplexes of at least two solute atoms. A dislocation segment after having surpassedan effective barrier sweeps in the subsequent elementary glide step an area containingcMn�cb solute atoms on the average.

Altogether, we arrive at �0 � �0��G0� cb�T � ��.

5.2.2 The hardening shear stress, �d

Detailed mechanical and TEM-studies have been performed on CuMn-crystals orientedalong [100] and [111] [5]. The hardening shear stress resulted as equal to the reducedstress, �d � �� 0�, and obeying the known relation [6]:

�d � �t Gb �1�2t � �1�

Here, �t is the average total dislocation density, G the shear modulus. A surprising re-sult 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 �t ishigher in an alloy than in the pure host. A further analysis showed that �t is storednearly completely in the cell walls, which are fully developed already at small stressesand strains. The next result of relevance was the increase of the wall area fraction fwwith solute concentration. Defining a mean dislocation density inside the cell walls, �w,by �w � �t�fw, we can rewrite Equation (1) as:

�d � �t f1�2w Gb �1�2

w � �w Gb �1�2w � �2�

The prefactor �w now turns out as independent of cMn and practically constant for afully developed cell structure. �w � 0�25 � 0�03 from the experiments favourably com-

5 On the Physical Parameters Governing the Flow Stress of Solid Solutions

92

pares with the lowest values for � calculated by theory [7] (cf. also [8]). This suggeststhe view that the most favourably oriented dislocation segments will cross the obstaclefield and will be followed via the unzipping effect by all others at nearly the samestress, which is the lowest possible one.

A TEM-investigation on Cu1.3 at% Mn crystals oriented for single glide [9], thefirst systematic TEM-study on a solid solution, yielded for the extended stage I a pre-vailing primary dislocation density, �prim, and a continuous decrease of the strain hard-ening rate with increasing strain. In stage II and above, the reduced flow stress wasfound as completely governed by the density of all secondary dislocations taken to-gether, �sec. It is:

�� 0� � �Gb �1�2sec � with � � 0�32 � 0�04 � �3�

The total shear stress, �, results as a linear superposition of a “solid solution stress”, �0,and a strain hardening stress, �d, as found also for the multiple glide crystals. It is com-pletely described by four parameters, apart from the obvious ones, T and ��:

� � �0��G0� cb�T� �� d��t� fw� T� �� 4�

The generalization for polycrystals adds the problem of compatibility of neighbouringgrains. It is of importance mainly for small stresses and strains and introduces essentiallythe average grain diameter as an additional parameter in the case of a random assembly ofgrains (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, ��DSA,increases proportional to the increase in the line concentration C of glide obstacles onarrested, “waiting” dislocations by �C during the waiting times, tw [11–13]. It is gener-ally assumed that �C is a function of �D�T�tw�, where D�T� is the diffusion coefficientof the underlying process with the activation energy Ea. The waiting time tw is connectedwith the strain rate �� via tw � ��, where � represents the “elementary strain” [14].

Phenomenologically, DSA can be described by an additional free activation en-thalpy �g � �g�Ea� tw� entering besides �G0 the well-known Arrhenius equation [15].

Ample DSA leads to flow stress instabilities (PLC-effect). Details of the processesinducing jerky flow can be studied in the region of stable flow preceding a PLC-regionby measuring with high accuracy i) stress-strain curves, �� (Figure 5.1), ii) strain ratesensitivities (SRS) of flow stress, �� ln ��, along whole stress-strain curves and overa wide range of temperature. The results are presented in the following.


93

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- andthree Al-concentrations have been studied, furthermore CuMn-crystals oriented for sin-gle and multiple ([100]) glide [15–20].

Figure 5.2 shows a survey of occurrence and types of instabilities inCu 2.1 at% Mn. The critical reduced stresses ��i � �0� (gained from about 40 ��-curves (see Figure 5.1) running parallel to the ��i � �0� axis) are plotted as function oftemperature T . There are three transitional temperature intervals, labelled �, � and �,where several regions of stable and unstable deformation alternate along ��-curves.Outside these intervals, the stress-strain curves are either stable or jerky throughout. Inthe small interval �, 290 �C�T�305 �C, an irregular sequence of bursts of type C


94

Figure 5.1: Schematic stress-strain curve showing the definition of the critical stresses ��i� andstrains ��i�. �0 is the critical flow stress (see [16]).

Figure 5.2: Mode of deformation map: dependence of reduced critical stresses ��i � �0� on tem-perature T for Cu 2.1 at% Mn. Basic strain rate ��1 � 2�45 � 10�6 s�1. The hatched areas representdomains of unstable deformation with the predominant types of instabilities indicated (see [18]).

prevents the existence of a unique dependence of �i on T (or ��) as could be establishedfor intervals � and �. For further details see [16, 18].

The strain hardening coefficient, which roughly remains constant for temperaturesup to about 300 K, decreases strongly with further increasing temperature owing to re-covery processes. At about T � 600 �C, it becomes nearly zero [18], the critical flowstress simultaneously vanishes as well as the additional enthalpy �g and a steady stateof deformation exists across the whole deformation curve. The solutes are now movingfreely through the lattice [21].

Figure 5.3 a gives stability maps for several Mn-concentrations over the intervals� and � 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 comparedwith those for a polycrystal of about 1.2 at% Mn. Contrarily, the boundary map for asingle glide ([sg]-) crystal of 2 at% Mn (Figure 5.4) looks quite different [19]. Only asingle boundary occurs over the whole range of temperatures. However, the curve canbe divided into two parts, which for good reasons are noted as intervals � and �, too(see Section 5.3.3).

Finally, boundary maps for CuMn-polycrystals have been compared with thosemeasured for CuAl [20]. Figure 5.5 presents characteristic examples. The complete cor-respondence of Cu 0.63 at% Mn with Cu 5 at% Al is obvious and indicates the exis-tence of two different PLC-domains. They are labelled as domains I and III. With in-


95

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


96

Figure 5.4: Reduced critical flow stresses ��x for Cu 2 at% Mn single glide crystals as functionsof 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 obtainedat different temperatures but constant basic strain rate ��1 � 2�5 � 10�6s�1. The reduced criticalstresses ��i � �0� indicate the transition between stable and unstable deformation (stress-straincurves 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]).

creasing concentration, a “bulge” develops on the boundary �2�T � ��. It is clearly visi-ble 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 boundedby the anomalous boundary �2

at the lower temperature side (Figure 5.5 b and e). Theisland of stability, which appears in Cu 1.1 at% Mn and Cu 2.1 at% Mn above about400 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 �i for the onset of jerky flow [16] on theone hand, and of changes in the flow stress, ��, 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 vari-ations 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, ��i, occur-ring immediately after a change in ��, and the stationary ones, ��s. (Remark: For singlecrystals, the flow stress � is always replaced by the resolved shear stress, �.)

It is the difference, ��s � ��i� � ��DSA�� which reflects the effect of DSAand causes a decrease of the SRS� �� ln ��T [19]. Analogously, for polycrystals isSRS� �� ln ��T [18].

The boundaries �2, �3 of the “island of stability” in temperature interval � (seeFigure 5.3) are governed by a thermally activated process as demonstrated in Figure5.7: A decrease in T is qualitatively equivalent to an increase in ��. We can take �i asindicating the onset of the thermally activated process and derive from

��i � �0� � A BT C ln �� ln �� A0 � B�C

T�5�

values for the activation energies Q� � B�C �� 2� 3� [18].


97

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

Another way of determining these energies starts from a consideration of the nor-malized 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�� alone for a polycrystal at various tempera-tures.

Following the above mentioned models [11, 12], the marked dependence ofSDSA � �Ss � Si� on T (and also ��) is for short enough tw described by the relation [18]:

�SDSA � cD�T��

� �n

� c��n exp�nEa

kT

� �� 6�

Here, Ea is the activation energy of pipe diffusion entering D�T� � D0 exp��Ea�kT�.Strain rate exponent n and Ea are best obtained from Equation (6) in regions, whereSDSA varies linearly with stress yielding constant slopes M � ��SDSA��T� ��. One eas-ily finds [18]:

n � � � lnM� ln ��

� �T� and nEa � � � lnM

��1�kT��

��

� �7�

The activation energies Q� and Ea�� found for the diffusion processes generating thePLC-domains I, II and III are compiled in Table 5.2 [20]. Where Q� and Ea�� can both


98

Figure 5.7: Dependence of the reduced critical stresses on (a) 1�T at ��1 � 2�45 � 10�6s�1, and on(b) ln ��1 at T � 460 K; both plots for Cu 2.1 at% Mn in interval � (see [18]).

be measured for the same process, they are found equal, Q� � Ea��, 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 asn � 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 2/3according to Cottrell and Bilby’s theory of lattice diffusion [23, 24]. Already the firstexperimental determination of n yielded, however, n � 1�3 and has been explained bya pipe-diffusion mechanism governing the DSA-process concerned [25].


99

Figure 5.8: Dimensionless instantaneous �Si� and stationary �Ss� SRS against reduced stress�� 0�� 0 � critical resolved shear stress; T � 263 K (see [22]).

Figure 5.9: The dependence of the stationary, normalized SRS on reduced stress forCu 3.5 at% Mn at temperatures of interval �. All data points refer to states of stable deformation.The critical stresses are indicated for 73.4 �C. The M� denote the linear slopes of the S-�� 0�-curves (see [18]).

Shortly after, Schlipf [26] pointed out that a more general relation than Equation(6) for SDSA is conceivable, viz. �SDSA � �Cq, which by use of �C � �D�T��ryields �SDSA � cq � ��qr (see also [27]). Now q � 1�2 and r � 2�3 would give an ex-ponent n � qr � 1�3 also in the case of lattice diffusion. On the other side, it becamemore 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 ofSDSA 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 thanthe van den Beukel model [11], and

• the data – taking the most reliable ones – are in favour of a simple proportionalityto solute concentration, i.e. q � 1.

This would exclude a lattice diffusion and is suggesting an own pipe diffusion mecha-nism for each DSA-process. Recent theoretical work [31] points to an even probableexistence of several modes of pipe diffusion [20] along dissociated dislocations.

A recently found method to measure immediately the average waiting time tw ofdislocations [32] showed that the elementary strain � continuously increases with theflow stress, the total increase never exceeding a factor of only 10. In principle, � is de-ducible from a knowledge of the dislocation arrangement ��t� �f � fw� and of the densityof glide barriers �cb� [19]. However, a general theory is still missing. Therefore, ��and with it tw, which governs stress transients, are still to be considered as parameters.


100

Table 5.2: Activation energies of DSA-processes in ��Al and ��Mn.

Average CuxAl CuxMn1

value5 10 0.63 1.3 2.1 3.5

�� [eV] 0.74 ±0.15 0.79 ±0.12 – – 0.88 ±0.05 0.86 ±0.10 2

�� [eV] × 0.75 ±0.10 × × × ×�� [eV] × 0.76 ±0.10 – 0.88 ±0.10 0.91 ±0.10 0.87 ±0.10�� [eV] 0.74 0.77 – 0.88 0.89 0.86

�� [eV] × × × 0.81 ±0.10 0.86 ±0.10 0.87 ±0.10

�� [eV] × 1.1 ±0.30 × × × ×

�� [eV] × 1.1 × 0.81 0.86 0.87

�� [eV] 1.42 ±0.25 × 1.9 1.53 ±0.10 1.27 ±0.10 1.15 ±0.10

�� [eV] 1.46 ±0.30 × – 1.-59±0.08 1.25 ±0.05 1.16 ±0.10

�� [eV] 1.44 × 1.9 1.56 1.26 1.15

1 Values for ��Mn from [18]; 2 �� 4.1 at% Mn; –: not measured; ×: not defined or not measur-able.

5.3Dynam

icStrain

Ageing

(DSA

)

101

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

Elementary process Characteristic magnitude Parameter (quanitatively measured) Literature

Solid solution hardening critical flow stress, ��(single crystal: crss, ��)

i) average activation enthalpy,�� (eV)ii) effective barrier concentr.��

Wille, Gieseke and Schwink [4];Neuhauser and Schwink [2]

Dislocation hardening reduced flow stress, ��

i) total dislocation density, ��[m–2]ii) volume fraction of disloc.walls, ��

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

Dynamic strain ageing (DSA) flow stress contribution, ��,or additional enthalpy,�� ,��

i) activation energy �� (eV),� � I, II, III, of the diffusioninducing DSAii) �� waiting timeof arrested dislocationsiii) strain rate exponent, �

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

Exhaustion of DSA limiting ��-value, ��

� ��

relaxation constant, � Springer, Nortmann and Schwink[30]

Transitions of DSA owing tovariations in ��

transition from instantaneousto stationary flow stress,��

relaxation time � waiting time�

Schwarz [13]; McCormick [34];Springer and Schwink [32]

Variations in mobile dislocationdensity, owing to variations in ��

active slip volume, � � � �� Schwink and Neuhauser [35];Neuhauser [36];Traub, Neuhauser and Schwink[37];Nortmann and Schwink [22]

The detailed analysis of the SRS also allows to evaluate quantitatively the varia-tions of the additional enthalpy by varying the strain rate, ��g�, along the wholestress-strain curves, and to determine the value of the quantity �g�� itself [30]. Itamounts up to about 10% of �G0 � 1�3 eV (see Section 5.3.1) [15, 30].

At higher flow stresses (�100 MPa), �g is observed to approach a limit with de-creasing strain rate. The course of �g�� can be best discussed for data on CuAl. Atthe time, the approach of the limit is described by a kind of relaxation parameter B. Itis concluded that an exhaustion of solute atoms available for the diffusion process inquestion limits the increase of ��DSA, and not a saturation of glide dislocations by thediffusion-induced glide obstacles [30].

Finally, the question has been addressed, whether the SRS might be influencedbesides of DSA-processes also by variations of the mobile dislocation density �m whenvarying the strain rate [22]. In fact, a superposition of both effects could be demon-strated. There exists a very small interval around a transition temperature, above whichDSA-effects are dominating the SRS, while �m-effects dominate below.

5.4 Summary and Relevancefor the Collaborative Research Centre

Shortly summarizing this report, we can say that any mechanism contributing to flowstress 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 experi-mental conditions, e.g. on temperature. In any case, the total number of relevant param-eters is defined and quite limited. Table 5.3 is to give a concise overview of all resultsobtained as far as they concern the parameters investigated.

The methods developed in this project to determine these parameters (cf. Table5.3) can be applied to any material. The parameters will enter any final constitutivematerial equations developed, e.g. those of project A6 of the Collaborative ResearchCentre (SFB). Results and experiences of our project have been also exchanged withproject A1. Throughout the work, there was an intimate contact to project A9.

References

[1] Ch. Schwink: Rev. Phys. Appl. 23 (1988) 395.[2] H. Neuhauser, 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.


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[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. Puschl, R. Frydman, G. Schock: phys. stat. sol. (a) 74 (1982) 211.[8] G. Saada: In: G. Thomas, J. Washburn (Eds.): Electron Microscopy and Strength of Crys-

tals. 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. Neuhauser: 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. Neuhauser: 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. Neuhauser: 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. Neuhauser: phys. stat. sol. 6 (1964) 679.[36] H. Neuhauser: In: F. R. N. Nabarro (Ed.): Dislocations in Solids, Vol. 6, North-Holland, Am-

sterdam, 1983, p. 319.[37] H. Traub, H. Neuhauser, 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 presentproject of the Collaborative Research Centre (SFB).

References

103

6 Inhomogeneity and Instability of Plastic Flowin Cu-Based Alloys

Hartmut Neuhauser*

6.1 Introduction

Slip deformation in crystal is inhomogeneous by nature as it is accomplished by theproduction and movement of dislocations on single crystallographic planes. Usually,only few dislocation sources are activated and produce slip on few planes, where, inparticular in fcc and hcp crystals and even more pronounced in alloys with low stack-ing-fault energy, many dislocations move on the same plane. This is provoked in partic-ular if the dislocations on their path through the crystal change the obstacle structure inthe slip plane, e.g. in short-range ordered or in short-range segregated alloys (i.e. innearly all alloys) so that the first dislocation feels a stronger “friction” than the suc-ceeding ones. As the macroscopic elongation of the sample is distributed in general het-erogeneously among the crystallographic planes, the quantities of resolved strain a andstrain rate �a, defined as:

a � l��0l0 and �a � �l��0l0 �1�

(with �l=macroscopic deformation rate, l0 =specimen length, �0 =Schmid orientation fac-tor), and commonly used in the formulation of constitutive equations, cannot be direct-ly connected with realistic dislocation behaviour.

Therefore, in this work, the local strain and strain rate in slip bands, which arethe active regions of the crystal [1], have been measured by a micro-cinematographicmethod [2]. Cu-based alloys turned out to be a convenient model system for experi-mental reasons: Single crystals can be grown easily in reasonable perfection and thestacking-fault energy � can be varied by changing the alloy composition. In Cu-2 . . . 16 at% Al, � varies from 35 to 5 mJ/m2 with increasing Al content, while it re-mains (nearly) constant (� � 40 mJ/m2) for Cu-2 . . . 17 at% Mn. Thus, the effects ofstacking-fault energy can be separated from those of solute hardening and short-rangeordering, which are comparable for both alloy systems.

104

* Technische Universitat Braunschweig, Institut fur Metallphysik und Nukleare Festkorperphysik,Mendelssohnstraße 3, D-38106 Braunschweig, Germany



While solid solution hardening has been extensively studied and is well docu-mented and appears well understood in the temperature range below room temperature[3–5], several open questions remain, which are particularly connected with inhomo-geneity of slip above ambient temperature. In a certain range of deformation condi-tions, even macroscopic deformation instabilities occur like the Portevin-Le Chatelier(PLC) effect. This effect appears to be a consequence of the mobility of solute atomsin 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 dur-ing deformation and accompanied by EM and AFM (atomic force microscope) investi-gations of the slip line fine structure and of dislocation structure by TEM. The conclu-sions reached so far as well as the still open questions are summarized. According tothe changes of principal mechanisms, the chapter will be divided into the rangesaround 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 per-formed in two special set-ups with tensile deformation machines equipped with light mi-croscopes. The slip steps are visualized in dark field illumination as bright lines, where thescattered light intensity is a measure of slip step height. The minimum step height resolvedis around Smin � 5 to 10 nm (depending on the quality of the electropolished crystal sur-face), changes of larger step heights down to �S � 5 nm can be resolved.

One apparatus is designed for very high resolution in time (down to 3 �s) [7, 8],using photo diodes and a storage oscilloscope with pretrigger parallel to video record-ing. From the rate of intensity increase and by comparison with interference micro-scopy of the same slip band after full development, the local rate of step height in-crease and thus the local shear rate can be determined. The time shift of curves of de-velopment recorded by two neighbouring photo diodes immediately yields the velocityof growth in length, corresponding to the velocity of screw dislocations if the observa-tions are performed on the “front” surface, where the plane with Burgers vector andcrystal axis cuts the crystal surface (cf. [9]). By using a second microscope and videosystem observing the opposite front surface of the plate-shaped crystal, the time, whichslip needs to traverse the crystal thickness, can be determined.

The second apparatus is designed for observations at various temperatures (up to500 �C) [10] and with a wide field of view between 0.3 and 4.2 mm in order to checkspatial correlations between activated slip bands. The video system usually records witha 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 lightmicroscopy, after deformation EM replica and AFM observations are performed. In ad-dition, in some cases, the dislocation structure developed during deformation steps hasbeen studied by TEM.

6.2 Some Experimental Details

105

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 arbi-trary constant stress values and recording strain and strain rate versus time. In particu-lar, the system permits rapid changes between deformation conditions, e.g. betweentesting at constant deformation rate and at constant stress. The specimen is inside a vac-uum tube (p<10–4 mbar) surrounded by a furnace reaching temperatures up to about1000 �C (±2 K). Load cell and extensometers (connected by rods to the grips) are situ-ated 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 dur-ing stress relaxation experiments has been measured for different alloy compositionsfor many single crystals oriented for single glide and for polycrystals. Crystals orientedfor 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) �0 and the ef-fective activation volumes V � kT�S (or strain rate sensitivities S � �� ln �a) deter-mined for Cu-Al single glide crystals, showing in Figure 6.1 a the typical low tempera-ture rise indicating thermal activation as rate controlling process, the plateau region atintermediate temperatures (now interpreted as a superposition of thermal activated glideand solute mobility, cf. Section 6.4) with a range of unstable deformation ending in amaximum of the crss, and the rapidly decreasing high temperature part (cf. Section6.5). These regions are also reflected in the strain rate sensitivity (Figure 6.1 b), whichwill 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. forCu-10 at% Al most frequent distances dsl =85 nm, step heights Ssl =25 nm), slip bandson the �m-scale (e.g. about dsb =5 �m in slip band bundles, 80 �m at the Luders bandfront, Ssb =120 nm), slip band bundles on the 100 �m-scale (e.g. average dbb =300 �m,integrated step height Sbb up to 15 �m) and up to the Luders band (e.g. widthBLB =3.8 mm, total shear SLB =36 �m) [11].

Direct measurements of the dislocation velocity from slip band growth in length(�xL) [12] result in

�s � �xL � 25 m�s �2�

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

106

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

�e � Ssb de�b � 3 m�s �3�


107

Figure 6.1: a) Temperature dependence of the critical resolved shear stress (crss �0) 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 serra-tions are plotted. b) Temperature dependence of the (normalized) strain rate sensitivity S � �� ln �a(determined from stationary back extrapolated stress jumps during strain rate changes) for one se-lected Al concentration (c=15 at%). Interval with arrows indicates PLC effect (jerky flow). Theplots 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).

a)

b)

results, where Ssb is the slope of the step height versus time curve in the very first fewms. The ratio �s/�e�8 (at least 3) appears reasonable in view of the interactionstrength of solute obstacles with different dislocation characters and with estimates offriction 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 thephoto diode; in Figure 6.2 b, the local shear rate during the development is shown in adouble log plot. After the very first rapid growth of step height, the rate slows downgradually when more and more slip lines are added to the slip band. While the veryfirst dislocation group appears to move under overstress, resulting in a local slip insta-bility 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. Thisbehaviour can be modelled by a (local) work hardening [11, 15]. It shows that the lo-cal strain rate

�aloc � �mb� �4�


108

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 ofnoise). b) Variation of the growth rate in step height Ssb (= local slopes of a)) for several slipbands in Cu-15 at% Al (compared with earlier results in Cu-30 at% Zn), plotted in double logscales versus time t. The dotted line indicates the growth rate, which would be necessary to ac-commodate the imposed deformation rate by one single slip band.

a)

b)

(�m =mobile dislocation density) varies with time in the activated slip zones by manyorders of magnitudes and that the assumption of an average strain rate according toOrowan’s equation (Equation (4)), which is frequently used in constitutive modelling,is not realistic and somewhat arbitrary. The local strain rate �aloc can be connected withthe external deformation rate only by using the “active crystal length” la instead of thetotal crystal length l0 in Equation (1):

�aloc � �l��0la � �5�

where

la � nabBsb �6�

(nab =number of simultaneously active slip bands, Bsb =active width of a slip band mea-sured along the crystal axis [2]) is a function of deformation rate, stress, strain, tem-perature, and time in general. Instead, the nucleation rate formulation of Orowan canbe 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 rateof slip band activations), and the details of slip band development do not matter be-cause 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 aslight stress drop in special experiments (using very thin, short specimens with the loadcell 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 (� 4 mm, length120 mm) in usual tensile machines with their large inertia.

The firstly activated dislocation source of a new slip band is always on that crys-tal surface, which due to the bending and lattice rotation by local shear feels a slightoverstress (surface “high”, see below). The average times tHL for the edge dislocationgroup to traverse the crystal from this front surface to the opposite one are found, forplate-shaped Cu-15 at% Al crystals of D=170 and 220 �m thickness, to be 11.1 and0.7 s, respectively, corresponding to average velocities of the edge dislocations of 22and 440 �m/s (slip plane inclined by 45� to the crystal axis). The large difference toEquation (3) is due to the opposing stress gradient along this dislocation path and re-flects the high strain rate sensitivity around room temperature (cf. Figure 6.1 b).

An important process in the formation of dislocation groups from each activatedsource is the partial destruction of obstacles by the dislocations cutting across obstaclesin the slip plane. In case of the present alloys Cu-Al and Cu-Mn, their well-known ten-dency to short-range ordering suggests that the effective obstacles in the yield regionare groups of solute atoms in an at least partially ordered configuration. This will bedestroyed by a cutting dislocation so that the next dislocation will be able to move at alower stress. Although some energetically favourable solute configurations will be “re-paired” by the following dislocations, the net effect is a destruction of “friction” to alower value in the activated slip plane. This process was modelled [14] using realistic


109

next-neighbour pair potentials determined from diffuse X-ray scattering measurementson Cu-15 at% Al crystals [18] in Monte Carlo simulations of a model crystal with themeasured short-range order, and the resulting dislocation configurations of the groupcompared with those observed in TEM [14] (Figure 6.3). This indicates quite high in-trinsic “friction” stresses of the original alloy. The dislocation group is able to move ata distinctly lower stress because the first dislocation feels, in addition to the externalstress, the internal stress from the following piled-up dislocations.

The resulting fluctuations in local stress are especially pronounced in the case ofCu-Al alloys, where the dislocation groups on single slip planes are much more ex-tended than in Cu-Mn alloys as a consequence of the low stacking-fault energy in theformer case, which prevents dislocations from easy cross-slip. This tendency is clearlyobserved 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 Fig-ure 6.4 c,d, and by AFM in Figure 6.4 e, f [19]. In particular, the high resolution of the


110

Figure 6.3: a) Variation of the diffuse antiphase boundary energy in the slip plane by passage of anumber n of dislocations crossing the slip plane and changing near neighbour short-range-orderedconfigurations. b) Interaction stresses between dislocations in single dislocation groups (�wwdotted lines) observed by TEM for annealed and quenched Cu-10.7 at% Al crystals. Full lines�SRO give the difference between these curves (�) and the simulation result (�) from a), assum-ing �SRO � �SRO�b [14, 23].

a)

b)

last method permits to decide that in Cu-Al, the activated slip line is indeed on a singlecrystallographic 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 forcross-slip (� � �Cu) appears to be the reason for the Cu-like slip line arrangement withclusters of activated slip planes during the work-hardening stages II and III, while inCu-Al alloys with its low � 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

a) b)

Figure 6.4: Comparison of dislocation structures: TEM micrographs after deformation in stage Iat 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.

c) d)

e) f)0.5 �m

diagrams may differ considerably from the local values relevant at the active disloca-tion sources and for the moving dislocations.

6.3.2 Development of slip band bundles and Luders band propagation

The process of successive activation of slip lines in the slip band and of slip bands in theslip band bundle or at the front of a propagating Luders band has been investigated indetail by observations on thin flat crystals [16] and by FEM calculations of the stressaround slip steps as well as calculations of the stress field resulting from excess disloca-tion groups below the surface necessary to shield the notch stress of the slip step (Figure6.5). These calculations show that in the surface region, maxima of shear stress occur incharacteristic distances ahead of a previously activated slip plane (irrespective of the de-tails of dislocation arrangement in the group), i.e. in a distance of 200 nm and in a distanceof 30 �m. The former corresponds to the observed distances of slip lines dse, the latter tothose of slip bands dsb; the numbers depend on the positions of the front and last disloca-tion of the excess group. Thus, the activation of a new source occurs under a certain over-stress, which explains the above-mentioned slip instability in the first stage of slip bandgrowth. It also indicates that the externally measured crss or yield stress in stage I hasto 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.


112

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

The above-mentioned differences between Cu-Al and Cu-Mn disappear in the meso-and macroscopic level: The appearance of slip bands, slip band bundles and the Ludersband is quite similar (Figure 6.6 a–d). In observations specially designed for examiningthe long-range correlations of slip by applying low magnification in the light micro-scope, it was found [23, 24] that the neat and simple Luders band configuration (Fig-ure 6.6 c,d) usually observed in thin flat crystals [16] can be produced also in thick cylin-drical crystals (4 mm �) if the external load (i.e. applied deformation rate) is selected lowenough. Such a deformation front, which is shown schematically in cross section in Fig-ure 6.7 (left side), propagates with a certain velocity �LB from one crystal grip to the otherduring tensile deformation in stage I (yield region) in a nearly stable configuration (solitarywave [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 oppositesurface “low” against a gradient of bending stress (Figure 6.5). The average plastic frontis normal to the crystal axis and the propagation velocity along the crystal, as determinedfrom the measured distances and times of front slip bands (Figure 6.8), is found to beproportional 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 (Fig-ure 6.6 a,b) whose trace across the crystal follows the crystallographic slip planes (Figure6.7, right side). Now, the stress due to the increased deformation rate appears to be highenough to activate sources more or less at random along the crystal length. They grow toslip band bundles by adding neighbouring slip bands according to the mechanism shown inFigure 6.5 (cf. [26, 27]). From such a slip band bundle, the Luders band starts when thebundle has reached a certain sufficiently high integrated shear, implying enough stressconcentration due to the bending moment, the thickness reduction and the lattice rota-


113

Figure 6.6: Light micrographs of slip band structure on the crystal front surface for the two defor-mation modes in stage I of Cu-Al and Cu-Mn: Formation and growth of slip band bundles: a) Cu-10.7 at% Al; b) Cu-17 at% Mn, and formation and propagation of a Luders band front: c) Cu-15 at% Al; d) Cu-17 at% Mn.

a) b)

c) d)


114

Figure 6.7: Schematic representation (crystal cross section along its axis) indicating the shear dis-tribution in the Luders band front which propagates with velocity �LB, and in slip band bundles(cf. Figure 6.6 a,b). The slip bands are initiated in the Luders band at surface “high”, at the edgeof the slip band bundle at its right on surface “high”, at its left on surface “low”, according to thebending stresses and the stress patterns of Figure 6.5.

Figure 6.8: Determination of Luders band propagation rates �LB from plots of cumulated dis-tances xF and times tF of the front slip bands of Luders bands at various external deformationrates of �l=2, 4, and 10 �m/s (selected below the critical value) for Cu-10.7 at% Al (a) and Cu-12 at% Al (b).

a)

b)

tion, which accompany the local shear of the single crystal (with slip planes inclined by 45�to the crystal axis). This stress concentration then helps to propagate the Luders band con-striction along the crystal. The Finite-Element Method (FEM) analysis of stresses (Figure6.9) shows a stress maximum at the tail and a minimum at the front of the Luders band; thelatter explains the large gaps between the front slip bands and indicate the necessity of localstress concentrations from neighbouring slip bands (Figure 6.5) to initiate the next new oneahead of the Luders band front. In a recent approximate treatment, Brechet et al. [28] havedescribed such transitions between homogeneous slip, bundled slip and propagating defor-mation fronts in quite general terms reflecting many of the above observations.

Macroscopically, the existence of stress concentrations is realized in the yieldpoints observed during first loading of the specimen. In fact, calculating the propaga-tion stress from the external load by using the specimen cross section at the most ac-tive part in the Luders band region, we arrive at the same stress as that is observed atthe yield point calculated from load and original cross section (cf. [23, 24]). This indi-cates 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

Figure 6.9: a) FEM analysis of the stress pattern around the Luders band front; b) Plot of the re-solved shear stress near the surface across the Luders band front from the sheared (left) to the vir-gin part (right), for different radii of curvature (R) in the Luders band region (cf. [23]). The in-creased stress at the left, mainly due to the reduced cross section, is compensated by work hard-ening (kinematic stress).

a)

b)

6.3.3 Comparison of single crystals and polycrystals

An important further stage of the investigations concerns the possibility to transfer thesingle crystal results to the case of polycrystals. As a first step in thin flat specimens ofCu-5, 10 and 15 at% Al with grain sizes around 200 �m, the slip bands have been ob-served during several steps of tensile deformation [30] recorded by video and examinedin detail after the deformation steps in the light and electron microscope. In exceed-ingly large grains, often fronts of slip bands propagate similar to slip band bundles orLuders bands in single crystals. In exceedingly small grains, slip activity is often re-tarded 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 indifferent parts of the grain (Figure 6.10). This reflects the local influence of compatibil-ity stresses exerted by the neighbouring grain. It also explains why not all, but mostslip systems are activated according to the magnitude of the Schmid factor. In the Cu-Al alloys, the plastic relaxation near grain boundaries occurs frequently, in spite of thelow stacking-fault energy, by cross-slipping of primary dislocations [12, 30]. This ap-pears to be easier than to activate new sources on secondary systems. It is important inparticular that the kinetics of single slip bands in polycrystals appear to be quite similar


116

Figure 6.10: Video records of slip line formation in single grains of a polycrystalline thin flat Cu-10 at% Al specimen shown at three stages of deformation (�=0.5, 1.5 and 8%) at room tempera-ture. The numbers in the scheme indicate the succession of activated slip planes.

to that in single crystals as shown in Table 6.1 for the average total times of activity ofsingle 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 Berg-mann [34] on a thin flat specimen of Cu-5 at% Al containing 3 grains of different knownorientations. The observations of slip band activity correspond well with the measure-ments 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 pro-ceeds by production and movement of groups of strongly correlated dislocations acrossslip zones. This strong correlation and the destruction of short-range order lead to local-ized deformation and micro-instabilities of slip. Owing to the variation of the slip ki-netics during the activity of each slip band, a description of the overall kinetics by thenucleation rates of slip bands (Equation (7)) including local work hardening (i.e. kine-matic stress) appears appropriate. Thus, the flow units used in [36] consist of such spa-tially and temporarily correlated dislocations in groups. Their local stress concentra-tions are important in the propagation of slip along the crystal. Details of the mecha-nisms and kinetics of dislocation multiplication inside the slip bands still remain to beexplored. The first steps done to study the influence of surrounding grains on the activ-ity of a considered grain in a polycrystal should be extended, in particular by combin-ing them with FEM analyses of the local stresses.


117

Table 6.1: Comparison of average times of formation of slip bands on single crystals (plateshaped, thickness 0.18 mm) and in grains of polycrystals (plate shaped, thickness 0.4 mm, grainsize 0.2 mm) for the Cu-Al alloys with 5, 10 and 15 at% Al. For the single crystals, the range ofobserved 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

6.4 Deformation Processes at Intermediate Temperatures

The range of “intermediate” temperatures is characterized by an increasing mobility of thesolute atoms in the alloy, in particular in the neighbourhood of dislocations. Although firstatomic site changes seem to occur in the dislocation core region already at temperatureswell below room temperature, as evidenced by strain ageing effects during and after stressrelaxation experiments [37], well pronounced influences of solute mobility are observed attemperatures exceeding room temperature (the lower, the higher the solute concentration,cf. [6]). In a certain range of temperature and external strain rate, dynamic strain ageingleads to repeated rapid local slip events even observable as serrations in the load-timecurve in ordinary deformation experiments, the well-documented Portevin-Le Chatelier(PLC) effect (e.g. [31]). Supplementing the research in [6], where most investigationsare performed in the range preceding this instability region, the present study concen-trates on the evolution of such plastic instabilities. Their temperature region for Cu-Alsingle crystals oriented for single glide and deformed in stage I is indicated in Fig-ure 6.1 a by the dotted lines. Figure 6.1 b shows that it nearly coincides with the rangeof negative strain rate sensitivity if this is determined from the back-extrapolated stresscourse during strain rate changes [38, 39]. For the more general behaviour and rangesof existence of the PLC effect during work hardening for various crystal orientationsand 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) si-multaneously with slip line recording by video, the course of PLC load drops has beendirectly 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 serrationgiven in Figure 6.11 b. Thus, in this range of temperature, one macroscopic instabilityevent 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 dislocationmultiplication occurs, where dislocations move fast enough to develop only minor so-lute clouds implying high dislocation mobility. The slip transfer mechanism of Sec-tion 6.3.2 (Figure 6.5) with local stresses in the surface region rapidly produces a seriesof neighbouring slip bands (i.e. a slip band bundle) at a rate higher than that necessaryto comply with the deformation rate imposed by the tensile machine. Therefore, theload decreases and thus the production rate and dislocation velocity, too. This in turnpermits the solute cloud to grow further and to slow down the dislocation until it stopssuddenly 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 inthe effective enthalpy ��G due to ageing:

�G � �G0 � ��G � �G0 � Kf �tw� �8�


118


119

Figure 6.11: Sequence of frames: a) with slip bands originating during a stress drop (b)), in the PLCregime (T=500 K) of a Cu-10 at% Al crystal deformed in stage I with a rate of �l � 2 � 10�3 mm/s.

a)

b)

in the waiting times for thermal activations

tw � tw0 exp ��G�kT� � �9�

where K and tw0 are constants and the function f describes the ageing kinetics. We find��G � 0.16 eV during the stress drop and �0.14 eV during reloading, i.e. �0.3 eVin total for Cu-10 at% Al at 580 K [39], which compares quite well with the values de-termined from different experiments and different arguments [42, 43]. This changeamounts to roughly 10% of the total effective activation enthalpy �G0 in this tempera-ture range of stress serrations.

The breakaway stress rises with temperature due to an increasing solute cloud, up toa stress maximum �0�TM� � �0M, which occurs at lower TM for higher solute concentra-tions c (Figure 6.1 a, in more detail Figure 6.12a). Beyond the crss maximum, no serra-tions occur and slip bands can no longer be detected: Slip becomes virtually homoge-neous for T � TM (cf. Section 6.5). The correlation of �0M with solute concentration (Fig-ure 6.12b) agrees quite well with the classical formula proposed by Friedel [44]:

�0M � A�W2m c�kTMb3� �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 byrapid 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 thedislocation, i.e. for c=2 . . . 15 at% Al: Wm � 0.12 eV taking the structure factor A=0.1as determined for Cu-Mn alloys by Endo et al. [45]. These Wm values compare wellwith earlier results from internal friction [46] and from theoretical estimates [47].

A summary of the temperature dependence of the correlation of serrations (load fluc-tuations) with slip activity is given in Figure 6.13, where, on the right, the mean stressdrop amplitude �� is plotted, while, on the left, the magnitude of simultaneously activeslip band bundles, naB, is given as determined from the video records according to:

naB � �l� �NbSsb � �11�

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


120

6.4.2 Analysis of stress-time series

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


121

Figure 6.12: a) High temperature part of the temperature dependence of the crss �0) (cf. Figure6.1 a) around its maximum, measured for various Al concentrations (2 . . . 15 at%) for Cu-Al singlecrystals oriented for single slip, at a deformation rate of �l=1.7 · 10–3 mm/s (crystal lengthl0 =100 mm) (� Cu-15 at% Al, � Cu-10 at% Al, � Cu-7.5 at% Al, n Cu-5 at% Al, � Cu-3.5 at% Al, s Cu-2 at% Al); b) plot of the maximum stress �0M =�0 (T=TM) at the temperaturesTM of the crss maxima versus alloy concentration to check Equation (10) by Friedel [44] for thetransition between dislocation breakaway from and dragging along of the solute cloud.

a)

b)

[48], which permits to take into account temporal correlations of the correlation inte-gral, such as in case of quasi-periodic behaviour, after checking the autocorrelationfunction (for determination of a proper cut-off) and the power spectrum (for detectingperiodicities).

In evaluations of stress-time series, special care must be taken in case of changes ofthe specimen structure during deformation as common in deformation due to work hard-ening. This is shown in the examples of Figure 6.14 [48] for polycrystals deformed in thePLC regime at different temperatures and for a single crystal oriented for single glide, bothfor 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 support-ing the recent theoretical treatment by Hahner [55]. For polycrystals, in certain ranges ofdeformation conditions at least some deterministic contributions are identified, which areperiodic and seem to correspond to the propagation of the various types of PLC bands. Thelong period “type A” serrations (at T=100 �C in Figure 6.14) is superposed by a shortperiod at higher temperature (“type B” at T=150 �C in Figure 6.14), while the single crys-tal 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 contin-uous propagation of plastic PLC deformation bands, type B corresponds to discontinuouspropagation of bands, and during type C, serrations at still higher temperature with spatially


122

Figure 6.13: Correlation of the temperature dependence of the number of active slip band bundlesnaB (Equation (11)) and the average height of stress serrations �� for Cu-10 at% Al crystals de-formed 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.

uncorrelated local deformation bands occur. Accordingly, for types A and B from an anal-ysis of the time series characteristic parameters of the deformation bands (band width, localplastic shear and shear rate in the band) and of their propagation rate �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 � (Figure 6.15a) and ��(Figure 6.15 b). The model of Jeanclaude and Fressengeas [57] would predict a decrease of�B with increasing � if spatial coupling of local deformations occurs by cross-slip, while anincrease would indicate spatial coupling by internal stresses [58] (cf. Figure 6.5). The ob-served dependence in Figure 6.15a then would mean a change of cross-slip transfer tointernal stress transfer with increasing temperature, which does not seem quite reason-able. Further investigations appear necessary and are under way for clarification.


123

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

Figure 6.15: a) Propagation rates �B of PLC deformation bands evaluated from time series likethose in Figure 6.14 for the serrations of type A (T=100 �C) and type B (T=150 �C) for varioustotal strains � at a strain rate of ��=1 ·10–4 s–1; b) strain dependence of type B propagation rates(T=150 �C) for variations of external strain rates �� l�l.

a) b)

6.4.3 Conclusion

The strain ageing process forming solute clouds around the dislocations leads to macro-scopically pronounced plastic instabilities in a certain range of deformation conditions,which are again intimately connected with strain localization. Here, the reason is thebreakaway of a dislocation from its solute cloud and subsequent rapid multiplication ofless aged dislocation groups. Thus, the overall kinetics (neglecting the serrations) canbe again described in the nucleation rate approach for aged dislocations, where the ki-netics of ageing enters the rate equations [40, 55, 56, 58]. The evolution of each singlestress instability event can be described in such an approach, too, while the details ofthe dislocation multiplication and in particular the role of cross-slip processes in theslip 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 occursin a nearly ideal homogeneous manner. This was checked by EM slip line replica andTEM: No traces of slip could be detected on the crystal surface, and TEM does notshow dislocation groups, but randomly distributed heavily jogged dislocations indicat-ing easy cross-slip of screw and climb of edge dislocations. Therefore, no slip line ob-servations are possible. In this range, viscous glide behaviour of dislocations can be as-sumed, and the classical Orowan equation (Equation (4)) �a � �mb� with a definite dis-location velocity � and mobile dislocation density �m appears realistic. According tothe analysis from Figure 6.12 b, these dislocations carry along their (unsaturated) solutecloud, which now decreases with increasing temperature for entropy reasons. Thus, thealloying effect diminishes with increasing temperature as seen in Figure 6.1 a and Fig-ure 6.12 a. The observation of a smaller yield stress for higher alloy concentrations (forc>5 at%) at T>TM, which looks surprising at first sight, can be explained by the well-known increase of the diffusion constant D (c) with solute concentration c [59] in thetreatment of Friedel [44]: The relation between strain rate �a and applied stress � is

�a � 2�m�b��D sinh ��b2��kT� � 2�mb3D��kT � �12�

where ��bcM�bc exp �Wm�kT� is the distance of pinning solute atoms along the dis-location. This relation also describes well the observed strain-rate sensitivity in stage Ifor T�TM (Figure 6.16c, where different c values are plotted), which agrees well if de-termined from either stress relaxations (Figure 6.16a) or from strain-rate changes (Fig-ure 6.16b), where the initial stress jumps (constant structure) are evaluated.

The course of stress relaxations in this temperature regime can be well describedby a viscous dislocation velocity [60]:


124


125

Figure 6.16: Strain-rate sensitivities (cf. Figure 6.1 b) in the range of elevated temperatures forsingle crystals of Cu-Al with different Al concentrations measured from stress relaxations (a) andfrom strain rate changes (b) taking the “initial” stress changes of the transients (i.e. without struc-tural changes) (��2 � 5��1� ��1 =1.7 · 10–5 s–1; symbols as in Figure 6.12 a); c) plot of the strain-ratesensitivity for T>TM (TM =temperature at the crss maxima in Figure 6.12 a) versus 1/T to checkEquation (12), for initial and stationary (i.e. back extrapolated) stress changes, using all data withdifferent alloy concentrations ≥5 at%.

a)

b)

c)

� � � � �13�

and either by a stress-dependent mobile dislocation density:

�m � �n � �14�

or by a Gaussian spectrum of free activation enthalpies. For the first approach, the ob-served n values correspond well with the m=n+1 values (=3.6 . . . 3.8± 0.5 for Cu-3.5 . . . 10 at% Al) usually found from creep experiments for this type of alloys [61]. Forthe latter approach, the temperature dependence of the average characteristic relaxationtimes changes abruptly at the temperature of the crss maximum, indicating again thechange of rate-controlling mechanism, i.e. breakaway of dislocations from their solutecloud for T<TM, solute diffusion in the non-saturated solute cloud dragged along withthe 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 Sec-tion 6.2. Figure 6.17 shows some typical creep curves in the plot of strain rate versusstrain: a) at a fixed stress for various temperatures, and b) at a fixed temperature for var-ious applied stresses for polycrystalline Cu-10 at% Al. After a rapid decrease, the strain-rate approaches stationarity (the following rapid increase of �� l�l is due to specimenconstriction and should be disregarded). In Figure 6.17b for sufficiently low stresses


126

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

a) b)

(or strain rates), creep occurs with an oscillating strain rate showing the characteristicsknown for dynamic recrystallization (e.g. [62]). For instance, the critical strain for the on-set of dynamic recrystallization increases in proportion to the applied stress (Fig-ure 6.18 a). The dynamic recrystallization can be induced by a rapid change to a lowerstress in the critical range (Figure 6.18b). This seems to be accompanied by changes ofthe dislocation structure, which are to be studied in more detail to obtain more informa-tion on the nature of the recovery processes in this temperature range T>TM.

The stationary creep rate, approximated by the minimum rate ��min in Figure 6.17,varies with stress and temperature (Figure 6.19) according to


127

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

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).

a) b)

��min � �5 exp ��Q�kT� �15�

with Q�2 eV, which approximates the activation for diffusion of solutes in the alloy orfor self diffusion. The stress exponent (m�5) is slightly higher than that quoted abovefrom stress relaxations, which has been clarified in [63, 64].

The first rapidly decreasing part of the creep curve contains information on dislo-cation multiplication. At very low stresses, this part of primary creep may even showincreasing strain rate for some time. Observed differences to creep tests in conventionalcreep machines [65] can be traced back to the different kinetics of loading. These pro-cesses will be explored further by rapid changes from strain rate to stress-controlledconditions 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 ki-netics can be well described by the viscous glide approach with the dislocation velocitygoverned by dragging of solute clouds and a stress-dependent mobile dislocation den-sity. This is the result of dislocation multiplication and simultaneous intensive recoveryprocesses, 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 co-workers, C. Engelke, A. Hampel, A. Nortmann, J. Plessing, in their dissertation works,and Ch. Achmus, U. Hoffmann, T. Kammler, M. Kugler, H. Rehfeld, S. Riedig, M.Schulke, H. Voss, G. Wenzel, A. Ziegenbein, in their diploma works.

In addition, I acknowledge gratefully the continuous discussions and cooperationwith Prof. Dr. Ch. Schwink, and the cooperation in SRO measurements with Prof. Dr.O. Scharpf (ILL Grenoble) and Dr. R. Caudron (LLB Saclay) by neutron scattering,and with Prof. Dr. G. Kostorz and Dr. B. Schonfeld (ETH Zurich) by X-ray scattering(with financial support of the Volkswagenstiftung). In particular, the financial supportof our work by the Deutsche Forschungsgemeinschaft in the Collaborative ResearchCentre (Sonderforschungsbereich, SFB 319-A9) is gratefully acknowledged.


128

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[35] E. Steck, J. Harder: Workshop on Large Plastic Deformations, Bad Honnef, 1994.[36] E. Steck: Int. J. Plasticity 1 (1985) 243–258.[37] H. Flor, H. Neuhauser: Acta Metall. 28 (1980) 939–948.[38] H. Neuhauser, J. Plessing, M. Schulke: J. Mech. Beh. Metals 2 (1990) 231–254.[39] C. Engelke, J. Plessing, H. Neuhauser: Mat. Sci. Eng. A 164 (1993) 235–239.[40] A. Kalk, A. Nortmann, Ch. Schwink: Phil. Mag. A 72 (1995) 315–339, 1239–1259.[41] J. Plessing: Dissertation TU Braunschweig, 1995.[42] F. Springer: Dissertation TU Braunschweig, 1994.[43] F. Springer, A. Nortmann, Ch. Schwink: Phys. Stat. Sol. (a) 170 (1998) 63–81.[44] J. Friedel: In: Dislocations, Pergamon, Oxford, 1964, pp. 405–468.[45] T. Endo, T. Shimada, G. Langdon: Acta Metall. 32 (1984) 1191–1199.[46] E.C. Oren, N.F. Fiore, C. L. Bauer: Acta Metall. 14 (1966) 245–250.[47] J. Saxl: Czech. J. Phys. B 14 (1964) 381–392.[48] C. Engelke: Dissertation TU Braunschweig, 1996.[49] M. Casdagli, S. Eubank, J.D. Farmer, J. Gibson: Physica D 51 (1991) 52–98.[50] P. Grassberger, I. Procaccia: Physica D 9 (1983) 189–208.[51] D.S. Broomhead, G.P. King: Physica D 20 (1986) 217–236.[52] M.T. Rosenstein, J. J. Collins, C. J. DeLuca: Physica D 65 (1993) 117–134.[53] J. Gao, Z. Zhang: Phys. Rev. E 49 (1994) 3807–3814.[54] G. Ananthakrishna, C. Fressengeas, M. Grosbras, J. Vergnol, C. Engelke, J. Plessing, H.

Neuhauser, E. Bouchaud, J. Planes, L.P. Kubin: Scr. Metall. Mater. 32 (1995) 1731–1737.[55] P. Hahner: Mater. Sci. Eng. A 164 (1993) 23–34; A 207 (1996) 208–215, 216–223.[56] P. G. McCormick: Trans. Ind. Inst. Metals 39 (1986) 98–106.[57] V. Jeanclaude, C. Fressengeas: Scripta Metall. Mater. 29 (1993) 1177–1182.[58] H. Neuhauser: In: E. Inan, K.Z. Markov (Eds.): Proc. 9. Int. Symp. on Continuum Models

and Discrete Systems, World Scientific, Singapore, 1998, pp. 491–502.[59] W. Jost: In: Diffusion, Verlag Steinkopff, Darmstadt, 1957, p. 109.[60] C. Engelke, P. Kruger, H. Neuhauser: Scripta Metall. Mater. 27 (1992) 371–376.[61] W. Blum: In: R.W. Cahn, P. Haasen, E. J. Kramer (Eds.): Materials Science and Technol-

ogy, Vol. 6: Deformation and Fracture (Vol.-Ed.: H. Mughrabi), VCH Weinheim, 1993,pp. 359–405.

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7 The Influence of Large Torsional Prestrainon the Texture Developmentand 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 andmore in use today. If the results are close to reality, the simulation can save costs in-volved in the forming of testing tools and shorten the development stage of new prod-ucts. But this aim can only be achieved if the model of the forming process is physi-cally plausible. The treatment of contact problems and the modelling of the material be-haviour, e. g., present many problems. The material properties of the anisotropy causedor at least modified by the forming process in particular are problematic.

In classical continuum mechanics, the material behaviour is described by phenom-enological 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 micro-scopic behaviour into account as in texture analysis (see Figure 7.1). Thus, it seemspossible to develop constitutive laws based on an improved physical basis and to usethem 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 ‘micro-scopic’ and ‘macroscopic’are often used. It is thus necessary to define the scale of ob-servation. The resolving power of the microscopic observer is usefully described by the

131

* Universitat Hannover, Institut fur Mechanik, Appelstraße 11, D-30167 Hannover, Germany



following definition. The observer knows the physical phenomenon and mechanism ofslipping, but he is not able to locate the area of slipping in the grain. Thus, if slippingoccurs, he is forced to treat it as a homogeneous in the grain-distributed action. Thisalso means that all points of the grain are describable by only one stress tensor or ve-locity gradient.

The expressions ‘macroscopic’, ‘global’ or ‘polycrystal’ are not related to a de-formed body of a special form; they refer to a volume of many crystals. This volume isoften called a control volume or representative volume, which is large compared withthe microscopic scale. Although it consists of many crystals, it is small in contrast toany deformed body. Thus, a deformed specimen consists of many representative vol-umes. To start calculations in the interior of the representative volume, one is forced tohave 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 incertain slip systems. A slip system consists of a slip direction and a slip plane. Theplanes and directions are determined by the structure of the crystal. In face-centred cu-bic 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 theshear stress � on the slip system exceeds a critical value �c. The deformation gradient,F and the velocity gradient L, relative to a lattice fixed frame, are then given by:

F � I � ��s� �mT�� and L � ��s� �mT

�� 1�

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

7 The Influence of Large Torsional Prestrain on the Texture Development

132

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

above are only valid if single slip occurs, but generally more than one slip system willbe operating simultaneously. The appropriate equations for multislip follow for the ve-locity gradient by superposition of single slips:

L ��

��s� �mT�� 2�

Nevertheless an analogue treatment of the deformation gradient F is not valid. Further-more, when the elastic distortion of the lattice is considered as well, the expressions be-come more complicated because the distorted lattice vectors must be used for an appro-priate 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 singlecrystal. It is more difficult to find a suitable averaging method to obtain the propertiesof the polycrystal. The interactions of the crystals at their grain boundaries during theirdeformation are so complex that there are still some simplifications necessary to makethe problem mathematically treatable.

Several texture models have been developed to deal with this problem. Some ofthem 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 basedon the Taylor model show better results compared with textures measured in experi-ments. It is therefore till now often the basis of texture simulations. Generally, mostmethods differ from each other in terms of whether the homogeneity of deformationsor the homogeneity of stresses are partly or completely satisfied. A comprehensiveoverview of modelling plastic deformation of polycrystals is given in [5].

All texture models require some basic data of the microscopic scale. Usually, atleast the following specifications are considered:

• The polycrystal consists of N� crystallites with equal volume. No restrictions aboutthe grain form are made.

• The orientation of each crystal is given by the Eulerian angles �1, � and �2. No in-formation 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 in-cluding their critical shear stresses. The assumption of the known critical shearstresses 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 texturemodel. The fundamental assumption of the model is that in each crystal, five slip sys-

7.2 The Model of Microscopic Structures

133

tems are activated in a way such that the microscopic velocity gradient for the incom-pressible flow is identical with the global one:

grad V � L � AT�

�5

��1

��s� �mT��

� �

��microscopic

A� �� with �� A� T� A� � �3�

The expression �� A� T� A� given by the transformation tensor A and its time deriva-

tive 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 setof 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. Espe-cially, if the critical shear stresses are equal on all slip systems, this selection criterionis not unique, and all combinations of five slip systems that lead to internal power in-side a tolerance limit are supposed to be active. As more than five slip systems operatesimultaneously, this is an extension of the Taylor theory. Another way to solve the am-biguous 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 thequality of the texture models.

If the magnitude of shear is known for all slip systems, the lattice spin is givenby:

�� W � AT�12

��1

��ges� �s� �mT

� �m� � sT��

� �A� � �4�

hence, A�� can be calculated. Integration of Equation (3) leads to the new orientation ofthe lattice. Finally, the microscopic stresses can be calculated with the known slip sys-tems. A macroscopic stress tensor and also a mean spin tensor are obtained by aver-aging the crystal data.

Normally the hardening of the crystal is considered by a law of the form:

��c� � f�

�12

��1

��

�� 1� � � � � 12� � �5�

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


134

��macroscopic

7.3 Initial Orientation Distributions

For a practical comparison of measured and calculated textures, the initial orientationsof the crystals should be measured as single orientations of single grains or as non-dis-cretized orientation distribution function (ODF). In theoretically based works and re-search projects, it is quite normal to start the calculation with an isotropic state. There-fore, 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 tocheck whether an initial isotropy is actually guaranteed and not only orthotropy. Manycriteria 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 de-viations from the isotropy configuration, there can be remarkable deviations of the elas-tic modulus for different directions of loading. In [6], the plastic isotropy is proved bycalculating the yield surfaces of the single crystals. If all these yield surfaces are regu-larly distributed in the stress plane, the distribution is thought to be isotropic. Thisapproach considers only the first possible slip system, and if multislip occurs, the iso-tropy 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 considerationof multislip.

A suitable test of the elastic isotropy is to calculate the average elastic stiffness ten-sor. The method introduced by Hill [8], which leads to good estimations in the case ofrandomly distributed crystals, seems to be the simplest and best method of approxima-tion. A quantity denoting the elastic anisotropy of an orientation distribution may be:

AE � E�max � E�min

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 tothe corresponding data of the single crystal. Thus, the value is independent of theconstants of the single crystal and only the quality of the distribution is assessed. In thecase of ideal isotropy, the quantity AE will vanish. A helpful visualization is to drawthe elastic properties of different directions as a body of elastic moduli. Here, distribu-tions with lower quality, concerning isotropy, show remarkable deviations from theideal spherical form.

The Taylor model is an ideal tool to check the plastic properties under considera-tion of multislip because at least five slip systems are active. The function

fAP�e� � 1 � AP�e� with AP�e� � �1��e� � �2��e� �c �7�

7.3 Initial Orientation Distributions

135

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

7.3.2 Strategies for isotropic distributions

A special strategy is only required if the distribution should consist of as few crystalsas possible. But given a later implementation of such a texture-based constitutive lawin a Finite-Element program, this should always be the aim.

Several authors use random distributions generated with the help of a randomgenerator. Unfortunately, these distributions are only usuable, considering the isotropy,if they consist of many orientations (>1000). Distributions created by a proper strategyare generally better than distributions generated randomly when the number of orienta-tions is equal.

Several strategies are based on a discretization of the Euler space. Here, the spacebuilt from the possible combinations of Euler angles is discretized. On account of thecrystal symmetry, it is not necessary to consider the entire Euler space; a small portionis 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 non-linear boundary and therefore it is not easy to discretize it. Muller [6] and Harren [7]use a corresponding discretization and obtain distributions of 32 � � �128 and 385 orien-tations, respectively. Isotropy is not satisfied in every case, but the distributions of Mul-ler [6] are better although they consist of fewer orientations.

Asaro and Needlemann [10] and Harren and Asaro [11] use a combination of spe-cific method and random distribution. The unit triangle of the stereographic projectionis used to fix one of the global axes. The attachment of the base in space is done withan angle given by a random generator. Figure 7.2 shows the elastic properties calcu-lated with data given in [11].

The distributions show a noticeable anisotropy, which is assumed to be caused bythe random generator. The method was used again to check the plastic isotropy withthe result that all distributions obtained had better properties than the original ones.

Another method is given by Muller [6], who discretized the surface of a sphere toobtain the positions of local basis vectors. This method leads to distributions of goodquality (see e.g. Figure 7.3), but it is always combined with the problem of the spheri-cal geometry.

This problem can be avoided if one takes the area of a circle for discretization andobtains the points on the sphere by an equal area projection. A detailed description of themethod is given in [12]. The quality of the distributions naturally depends on the divisionof the area and on the number of orientations, but for the same number of orientations, theisotropy is better or at least comparable to that of the so-called Kugel distributions. Fig-ure 7.4 shows the isotropy test of a distribution generated with this method. Although itconsists of only roughly one hundred orientations, the isotropy is nearly guaranteed.


136

7.4 Numerical Calculation of Yield Surfaces

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


137

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].

procedure of averaging and the basic assumptions vary. The methods can be catego-rized as follows:

Static methods of averaging are based only on the Schmid law and no strain isconsidered. Methods of this type are not suitable for a comparison with experimentaldata, as the later are usually measured with an offset strain. In [6], a method is pro-posed based on averaging the single crystal yield surfaces. This method can also con-sider kinematic hardening when the surface lies outside the origin. Figure 7.5 showstwo yield surfaces on an initial distribution. The values of the stresses �XX, �YY and�XY are related to a mean value of the critical shear stresses. This normalization is alsodone in the figures below. In [13], this method is combined with some offset simula-tions. 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 Kovacs[14]. This method uses a double averaging, first in the grain and second for the poly-crystal. It is shown in [12] that the calculated yield stress is the harmonic mean of thefive lowest possible stresses causing yielding in different slip systems. The arithmetricor geometric mean may be used in the same way.

The classical Taylor yield surfaces are based on statics as well. The yield surfacesshown in Figure 7.6 have been calculated by applying 80 loading paths, which aremarked by the arrows. The best and fastest method for calculating the yield surface inthis manner was introduced by Bunge [15].

Stress-controlled methods are based on a global given stress tensor. The stress isincreased incrementally until the shear stress in one slip system exceeds the critical val-ue. It is then possible to calculate the amount of shearing that is needed for a static equi-librium with the hardening law. If the critical shear stress is not reached during a step, thedeformation is assumed to be purely elastic. After all deformations of the crystals havebeen obtained, the mean value of strain is calculated. The method continues until the off-set strain is reached. Unfortunately, only the stress is given and no information about theglobal velocity gradient is supplied. Therefore, antimetric parts are hardly considered. Butfor small deformations (e. g. for a simulation of the elastic-plastic transition), this methodmay be suitable. Figure 7.7 shows initial yield surfaces of an initial kugel distribution. Thegraph is due to an ideal calculation, and the symbols correspond to a calculation underconsideration of strain hardening, loading path effects and orientation alterations. The off-set strain used is 0.2%, which is a standard value in material testing. One may notice that


138

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

although a large offset is used, the resultant yield surfaces are smaller than the ones cal-culated with the Taylor model. Therefore, the later ones are only valid for comparison withyield 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 pathlike an experiment. When the components of the global velocity gradient are given forthe stress plane considered, e.g. in the form

Lij � fXLXij � fYL

Yij with fX � cos � and fY � sin � � �8�

it is possible to apply the loading path desired by choosing suitable values for the an-gle �. LX

ij and LYij are the tensor components of pure loading in X and Y direction, re-

spectively. When � changes, the equivalent strain rate changes, too; it is therefore nec-essary to vary the time increment of the integration to achieve a constant step ofequivalent strain increment during each loading step. Thus, this method is suitable forcalculating yield surfaces as shown in Figure 7.8. Exactly 80 loading paths startingwith pure tension and then continuing counter clockwise round the stress plane are ap-plied. For comparison, the ideal Taylor yield surfaces are shown too. This model showsthe expected effect as the expanding of the yield surfaces caused by the loading pathunder consideration of hardening. The hardening law used is the isotropic PAN lawwith parameters proposed in [6] and the offset strain is 0.1%.


139

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

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

When kinematic hardening is considered, the yield surfaces become distorted andmay not be closed.

The only disadvantage is that the classical Taylor theory starts with the full plasticmaterial state. Thus, the elastic-plastic transition is not taken into consideration. A bet-ter method might be the Lin model [16], which similar to the Taylor model assumesthat all crystals have the same strain. Furthermore, nearly the same hardening law canbe used. This method is used in [6] for the calculation of offset-strain dependent yieldsurfaces. The disadvantage is the small deformation area of application. Thus, it is notuseful for texture simulations. The problem is discussed further in [12], and it is shownthat 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 pre-strained materials. The large deformation was achieved with a torsion-testing machineat the Institut fur Mechanik of the Universitat Hannover. The measurement of the yieldsurfaces was done with a testing machine at the Institut fur Stahlbau of the TechnischeUniversitat Braunschweig. The material of the specimens was always the aluminium al-loy AlMg3.

7.5.1 Prestraining of the specimens

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


140

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

to ensure that the cylindrical form of the specimens was maintained, a lubricated man-drel was inserted inside the specimens. A maximal amount of shear of�a � tan � 1�5 could be reached with that configuration, where describes the angleof an axial direction on the surface of the specimen after the deformation. That meansa twist of about 360 degrees for the specimens. The elongation of the specimen wasnot suppressed with the result that an elongation always occurred, which nearly de-pended linearly from the twisting angle in agreement with the research done by Poh-landt [17] with specimens of aluminium. The maximal elongation was �� 1�25 mm.Since the measurement of the yield surfaces was done in another apparatus, the speci-mens were fully unloaded after the torsional deformation.

7.5.2 Yield-surface measurement

Four different material states have been investigated: specimens without any prestrain andones with �a =0.5, �a =1.0 and �a =1.5 magnitudes of shear. The testing apparatus was astrain-controlled machine with the capability of combined tension-torsion loadings. Theyield point of the material was detected with the offset strain definition. In all tests, onespecimen was used for 16 loading paths, starting with pure tension and then continuingcounter clockwise round the �-�-plane. This was done for three reasons. First, this re-duced the costs of specimens. Second, it was not guaranteed that the prestrain is reprodu-cible and last, the multipath measurement data are needed for the comparison with thetheoretical models. These data are ideal to check whether the texture model includingthe 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 ofthe total strain increment in an elastic and a plastic part. If the total strain increment isgiven by the testing machine, the plastic parts of it are given by:

�pl � �ges � ��

Eand ��pl � ��ges �

��

G�9�

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

�vMpl �

��2

pl �13��2

pl

�� 10�

The yielding point was reached when the calculated plastic strain exceeded the givenoffset strain:

��vM

pl � off � �11�

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


141

errors may be the result. If E is measured too large, the resultant yield stress will besmaller than the real one. In an extreme case, yielding is supposed although thematerial is still in the elastic state. On the other hand, if E is measured too small, theresultant yield stress will be larger than the real one. The shear modulus G has an ap-propriate influence. This possible error in determining the yield stress increases withdecreasing offset. Data obtained by using very small offsets should therefore be treatedwith caution.

The determination of the elastic constants E and G is normally done with the firstmeasured points of a new loading path. The best way is to calculate the regressioncoefficients. Although the regression coefficient in the elastic range should always beconstant, that is practically not the case. It always varies in a small range depending onthe number of considered measuring points as shown in Figure 7.9.

Thus, if another number of measuring points is selected for calculating the modu-lus E, the resultant value E and as a consequence also the resultant yield stress will bechanged. This is an especially critical case for the modulus E; the shear modulus Gshows 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 isgiven when the maximum yield stress in the tensile component of stress is altered bythe strain hardening in torsion and vice versa. Normally, the cross-effect and related is-sues are investigated by the measurement of yield surfaces when the plastic deforma-tion at most reaches the usually small offset strain. This tensile test was realized to in-vestigate 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 cylindri-cal specimen was twisted up to fracture, which occurs at a shear rate of �a =1.65. The


142

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

tensile specimen was produced from the broken rest as shown in Figure 7.10. An esti-mate led to an amount of shear of �a =0.55 at the radius of the final test specimen, butas a result of the processing, the two specimens were not distinguishable. The result ofthe 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, themechanical strength properties are greater than those of the unstrained specimen. Val-ues never reachable for the unstrained specimen were obtained. This shows that there isa remarkable cross-effect.

7.5.4 Measured yield surfaces

Some measured yield surfaces are presented below. Detailed discussions and further in-vestigations 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 forthe unstrained specimens, while this value was too small for the prestrained specimens.There were several runaways among the data measured and therefore the smallest off-set 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 in-creasing offset.

As expected, the yield surfaces have an elliptical form and for small values, theaxial ratio �/� is closely to the von Mises yield surface. This ratio, however, increaseswith increasing offset as well. The data obtained from the largest offset used show anexpansion of the surface, which is surely caused by the specific loading path and thelarge offset of 0.1% inducing significant plastic deformation.


143

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

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 pre-

strained specimens. This is causually connected with the loading path because in thesecond and third quadrant, no such areas occur. On the other hand, this concave area isdue to the first measured point and, therefore, it is the first loading differing from thetorsional preloading. This might be an important fact. Furthermore, the surfaces show ahardening with increasing degree of prestrain. There is a significant distortion, a flatten-ing of the portion of the surface opposite the loading direction, and a kinematic harden-ing 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 direc-tion. Furthermore, a hardening with increasing degree of prestraining is noticeable. Ex-ceptionally, the yield surface of the unstrained specimen is measured with an alternateloading path, which does not affect the shape strongly.


144

Figure 7.11: Yield surfaces of unstrained specimens.

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

Figures 7.12 and 7.13 show that it is often difficult to assign the characteristicsobserved. One may ask if the effects are due to the prestraining or to the parameters ofthe measurement. Especially, the loading path for each specimen could be the cause ofsome effects. In order to investigate the influence of these parameters, some specimenswere 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 un-strained material are shown in Figure 7.14, where only the surfaces measured with thesmall offset are presented. The data measured characterized by the * is of the thirdmeasurement of this specimen. Thus, an influence of the loading path can be seen be-cause this yield surface is slightly shifted to the direction of the last loading of the pre-vious path with the large offset.

In fact, there is an influence of the loading path, which does not seem to be toolarge because the form of the surface is not affected.

Surprisingly, the prestrained material shows a very different behaviour. In Fig-ure 7.15, correspondent measurement of a specimen prestrained up to �a =0.5 is shown.


145

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

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

The first surface measured with the small offset shows the properties already de-tected in Figure 7.12. The second has nearly an elliptical form. The third surface, mea-sured with the small offset, is a slightly shifted ellipse. Considering the previous yieldsurface, 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 re-markably. Thus, the prestrained specimens are very sensitive to further deformationscompared with the unstrained ones.

7.5.5 Discussion of the results

The measurement of yield surfaces is not problematic for unstrained specimens evenwhen 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 formof the yield surface is not known, it is difficult to identify runaway data and to assignthe effects to parameters of the measure procedure. An amplification of these effects isdue to the problem of determination of the correct elastic modulus.

In comparison with similar investigations (e.g. in [18–24]) agreement as well asdifferent results can be found.

7.6 Conclusion

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


146

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

distortion and the resultant anisotropy are often assumed to be caused by the orienta-tion distribution of the single crystals in the material.

The results presented prove that especially prestrained material is very sensitive tosmall deformation. This means that the principle form of the yield surface is stronglysensitive to small plastic deformations. Since this small deformation hardly affects thetexture of the material, it must be assumed that the texture is not the real cause for thedistortion of the yield surface. Additional events and mechanisms must occur in thematerial during any plastic deformation.

Further investigations on the numerical calculation of yield surfaces will be under-taken. Especially the question concerning, which method leads to results similar to thesurfaces measured and what kind of microscopic hardening law is needed, will be con-sidered. The fact that almost all parameters of the hardening law must be identified bythe mechanical properties of the polycrystal is problematic. At least, one should be ableto identify all these parameters with standard methods in material testing. Otherwise, itdoes not make sense to use microscopic-based material laws. The final aim is to do thecalculation first and then proceed in manufacturing. The other way of doing an experi-ment first and then trying to reach the same results in simulation may be practicablefor research projects, but this is surely not senseful for practical applications.

Finally, a search for a texture model to describe small deformations as well aslarge 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, Cam-bridge, 1992.

[2] G. Sachs: Zur Ableitung einer Fließbedingung. Zeitschrift des Vereins deutscher Ingenieure72 (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 ofMaterials Science and Technology: A Comprehensive Treatment (Vol.-Eds.: R. W. Cahn, P.Haasen, E. J. Kramer), VCH, Weinheim, 1993, pp. 89–136.

[6] M. Muller: Plastische Anisotropie polykristalliner Materialien als Folge der Texturentwick-lung. VDI Fortschrittsberichte Reihe 11: Mechanik/Bruchmechanik, VDI-Verlag, Dussel-dorf, 1993.

[7] S.V. Harren: The Finite Deformation of Rate-Dependent Polycrystals: I. A Self-ConsistentFramework. 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.

References

147

[10] R. J. Asaro, A. Needlemann: Texture Development and Strain Hardening in Rate DependentPolycrystals. Acta. metall. 33 (1985) 923–953.

[11] S.V. Harren, R. J. Asaro: Nonuniform Deformations in Polycrystals and the Aspects of theValidity of the Taylor Model. J. Mech. Phys. Solids 37 (1989) 191–232.

[12] N. Wellerdick-Wojtasik: Theoretische und experimentelle Untersuchungen zur Fließflachen-entwicklung bei großen Scherdeformationen. Dissertation Universitat Hannover, 1997.

[13] D. Besdo, M. Muller: The Influence of Texture Development on the Plastic Behaviour ofPolycrystals. In: D. Besdo, E. Stein (Eds.): Finite Inelastic Deformations – Theory and Ap-plications. IUTAM Symposium Hannover/Germany 1991, Springer-Verlag, Berlin, Heidel-berg, 1992, pp. 135–144.

[14] L.S. Toth, I. Kovacs: A New Method for Calculation of the Plastic Properties of Fibre Tex-tures 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, Gottingen, 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. Pohlandt: Beitrag zur Optimierung der Probengestalt und zur Auswertung des Torsions-

versuches. 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, Interna-tional Seminar on Multiaxial Plasticity, 1992, pp. 1–17.

[24] A.S. Khan, X. Wang: An Experimental Study on Subsequent Yield Surface after FiniteShear Prestraining. Int. J. of Plasticity 9 (1993) 889–905.


148

8 Parameter Identification of Inelastic DeformationLaws 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 bemeasured directly.

The classical method of the determination of material parameters is to demand aquite good agreement between measured data from properly chosen experiments andcomparative data taken from numerical analysis. This will be carried out by the optimi-zation of a least-squares functional.

Furtherly, the parameter identification based on experiments with inhomogeneousstress-strain fields, the usage of global and local comparative quantities in the objectivefunction and optimization by deterministic methods will be described.

8.2 General Procedure

In addition to classical material-testing methods, current research is done to identifymaterial parameters of inelastic deformation laws by the experimental and theoreticalanalysis of inhomogeneous strain and stress fields. A new method is the parameteridentification using the comparison of numerical results obtained by the Finite-Element

149

* Technische Universitat Chemnitz, Institut fur Mechanik, Straße der Nationen 62,D-09009 Chemnitz, Germany



method with experimental data, for instance, with displacement fields measured by op-tical techniques [1–7]. The papers [1–4] were realized within the Collaborative Re-search Centre (Sonderforschungsbereich 319).

Unlike this Finite-Element algorithm based method, in this paper, another proce-dure is presented to identify material parameters of inelastic deformation laws. Theprinciple consists in the experimental determination of the strain distributions in the li-gament of a notched bending specimen at several load steps and the numerical integra-tion of the deformation law at a certain number of points along the ligament with mea-sured strain increments as load. The actual material parameters can be found using theglobal equilibrium of the stresses integrated along the ligament with the known exter-nal loads. Besides also local quantities, for instance, the stresses in the grooves of thenotch 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 andmaterials as sheet metals or metal plates are studied. The elastic properties should beisotropic. Viscoplastic effects are neglected. An initial anisotropy, especially a planarorthotropy 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 asused in the material subroutines of the integration algorithm (cf. Section 8.6.1) will beconsidered. At the yield limit holds the yield condition:

F��h� p� � 0 � �1�

The linear elasticity law

�� E�� 2�

is valid for loads in the elastic domain

F � 0 or F � 0 and ��T �F��

� ��TET �F��

� 0 � �3�

For loads into the plastic domain

F � 0 and ��T �F��

� ��TCT �F��

� 0 � �4�

the deformation law becomes:

�� el � ��pl � E�1 �� F��

� �5�

8 Parameter Identification of Inelastic Deformation Laws

150

In this case, the inner variables are assumed to develop in accordance with:

�h � ��q�� h� p� � �6�

The material stiffness matrix C��h� p� arises from Equation (5) by elimination of ��with the help of the consistency condition.

8.2 The Deformation Law of Inelastic Solids

151

Figure 8.1: Scheme for the identification of material parameters by bending tests.

8.4 Bending of Rectangular Beams

8.4.1 Principle

The simultaneous determination of the uniaxial stress-strain curves for tension and com-pression by bending test is known since 1910 [8]. On the other hand, examples for ap-plication were relatively rare publicated [9–11]. Here, this technique is applied to ana-lyse the material properties in the delivery state. Besides, it was developed to calculatethe initial yield surface (Figure 8.2).

8.4.2 Experimental technique

Test material for the own investigation was the stainless steel X6CrNiTi 18.10, whichexhibits a gradual transition from the elastic to the plastic behaviour. All specimenswere made from one and the same metal plate with a thickness of 6 mm.

It is assumed that the sheet metal has a planar orthotropy coming from the produc-tion process. To determine these initially anisotropic material properties, two specimensare prepared, the axes of which are parallel to the orthotropy directions (Figure 8.3).

Bending tests, being able to provide yield curves and the initial yield surface,were carried out in a specially constructed four-point bending device positioned in aconventional 100 kN material testing machine. The longitudinal forces in the specimenscan be neglected because the four loads are easily moveable in horizontal directionwith the increasing deformation (Figure 8.4).


152

Figure 8.2: General procedure for the investigation of initial material properties.

All tests were made at a velocity of 0.108 mm/min and at a maximum strain rateof about 10–5 s–1.

The axial and lateral strains at the outer fibres of the bending beams were mea-sured with high-elongation strain gages up to maximum strains of 5%. In separate ex-tensive investigations, the measuring accuracy of strain gages was analysed at higherstrains using the Moire technique as an experimental reference method. As result, equa-


153

Figure 8.3: Specimen geometry for bending beams.

Figure 8.4: Device for four-point-bending.

tions for the determination of strain in the elastic-plastic region were obtained, whereasin the elastic region, the producer’s strain gage factor can be used [12, 13].

As an example for the primary experimental results, the measured curves for thebending moment and the strains at the outer fibres are shown in Figure 8.5.

The high density of data and the smoothness of the curves are a good basis forevaluation. A maximum strain level of about 0.5% is sufficient to determine initialmaterial properties and to study the elastic-plastic transition.


154

Figure 8.5: Experimental results in pure bending of a rectangular beam: a) bending moment; b)strains at the outer fibres (C compression, T tension).

8.4.3 Evaluation

8.4.3.1 Determination of the yield curves

The following fundamental assumptions are made for the evaluation of elastic-plasticpure bending tests:

• The cross-sectional areas remain plane.• The stress state is uniaxial.

The trapezoid-shaped distortion of the cross-section is taken into account, but the defor-mations should be only moderate so that the stress states can be assumed as uniaxial.

In pure bending, the distribution of the usual technical strain � � �l� l0��l0 is alinear function of the coordinate y at the deformed beam (Figure 8.6):

�x�y� � ��y � � 1h��T

x � �Cx � �7�

with � as the curvature of the neutral axis. Then, the bending stress curve is an imageof the uniaxial stress-strain relation of the material (Figure 8.6).

The equivalence between the bending stresses �x and the bending moment Mb

and the longitudinal force N � 0, respectively, requires:

Mb�� hC��

�hT��

�x�y�b�y�ydy � 1�2

��Tx ��

�Cx ��

�x��x�b��x��xd�x � �8�

N�� hC��

�hT��

�x�y�b�y�dy � � 1�

��Tx ��

�Cx ��

�x��x�b��x�d�x � 0 � �9�


155

Figure 8.6: Strain and stress in a rectangular beam under pure bending.

In Equations (8) and (9), �x��x� represents the uniaxial yield curves for tension andcompression. The actual width b of the beam at the coordinate y can be transformed inb��x� � b0�1 � �z��x�� with b0 as the initial width of the rectangular cross-section.

Differentiation of Equations (8) and (9) with respect to the parameter � gives:

�1 � �Tz ��T

xd�T

x

d��T

x � �1 � �Cz ��C

xd�C

x

d��C

x � �

b02M � �

dMd�

� �� 10�

�1 � �Tz �

d�Tx

d��T

x � �1 � �Cz �

d�Cx

d��C

x � 0 � �11�

and from there, the stresses �Tx ��T

x � and �Cx ��C

x � in the outer fibres:

�T�Cx � 1

b0h0

2M � �dMd�

d�T�Cx

d�

1hh0

�1 � �T�Cz �

�12�

depending on the curvature �. In Equation (12), �Tx �� and �C

x �� represent the corre-sponding strains in the outer fibres of the bending beam.

In contrast to the former works [8–11], the last term in Equation (12) reflects thetrapezoid-shaped distortion of the cross-section and takes into account anisotropic, espe-cially orthotropic material behaviour. The quantity h�h0 in this term can be determined by:

hh0

� �Tx � �C

x

�

�hC

�hT

dy1 � �y�y�

� �Tx � �C

x��T

�C

�1 � �x��1 � �z��x��d�x

�13�

assuming incompressibility for the total strains.In Equation (12), the denominator causes different yield curves for tension and

compression, respectively, whereas the numerator defines the general level of the stress.These two influences are already seen in the primary experimental results of Figure 8.5.

As a first step in evaluation, the limit of the linear part in the curves of Figure 8.5was estimated by a multiphase regression method. From these data, the elasticconstants in form of the Young’s modulus E and the Poisson’s ratio � can be easily de-termined using linear elasticity. In a second step, the data curves were approximated bycubic spline functions. After differentiating, the stresses can be calculated numericallypoint by point from Equation (12).

In Figure 8.7, stress-strain relations following from the data of Figure 8.5 are shown.In addition to Figure 8.5, the yield curves for the whole strain range up to 5% are

presented in Figure 8.8.The initial yield locus curve is characterized by the uniaxial yield stresses and the

directions of d�pl in these stress points (cf. Figure 8.2). The yield stresses in tensionand compression, respectively, are defined from the stress-strain relations assuming arelatively small offset strain of 0.1‰. The plastic strains can be calculated from the


156


157

Figure 8.7: Yield curves determined from the data of Figure 8.5.

Figure 8.8: Yield curves up to a maximum strain of 5%.

measured total strains at the outer fibres in longitudinal and transverse direction usingthe incompressibility of plastic deformation and the elastic deformation law. Then, thedirection of d�pl can be found from differentiating the curves �pl

y ��plx � at approximately

�plx � 0�1‰ (Figure 8.9).

8.4.3.2 Determination of the initial yield-locus curve

From bending tests of straight specimens, the yield stresses and the directions of plasticflow, i.e. the normal directions of the yield surface at measured yield stresses, for puretension and compression in x- and y-direction were determined.

If one assumes a quadratic yield function, the most general expression for princi-pal stress states is the quadratic form:

f ��x� �y� � h1�2x � h2�x�y � h3�

2y � h4�x � h5�y � h6 � 0 � �14�

One constant is free, hence h2 was set to be (–1) and h6 will be positive for:

f ��x� �y� � h1�2x � �x�y � h3�

2y � h4�x � h5�y � h6 � 0 � �15�

The first part of the objective function should minimize the squares of the yield func-tion at the measured yield stresses ��:


158

Figure 8.9: Plastic strain �ply ��pl

x � from the data of Figure 8.5.

�1 � 12

�q

f ��x�q� � ��y�q� �� 2

� min � �16�

Then, the optimality condition

grad �1 � 0 �17�

gives a system of linear equations for the constants h1, h3, h4, h5 and h6:

��4x� ��2

x��2y� ��3

x� ��2x��y� ��2

x��2

x��2y� ��4

y� ��x��2y� ��3

y� ��2y�

��3x� ��x��

2y� ��2

x� ��x��y� ��x��2

x��y� ��3y� ��x��y� ��2

y� ��y��2

x� ��2y� ��x� ��y� �1�

��

��

h1

h3

h4

h5

h6

��

��

��3x��y�

��x��3y�

��2x��y�

��x��2y�

��x��y�

��

�� 18�

with �� q�� . All stresses are normalized.

For yield stresses, all representing only pure tension and compression, the rightside will be zero. To overcome this, the normal directions n � �nx� ny� were taken intoa second part of the objective function:

�2 � 12

�q

�nx�q��F

��y�q�� ny�q�

�F��x�q�

� �2

� min � �19�

The optimality condition gives another system of linear equations for h1, h3, h4 and h5:

�4��2x�n

2y� ��4��x��y�nx�ny� �2��x�n

2y� ��2��x�nz�ny�

��4��x��y�nx�ny� �4��2y�n

2x� ��2��y�nx�ny� �2��y�n

2x�

�2��x�n2y� ��2��y�nx�ny� ��n2

y� ��nx�ny��2��x�nx�ny� �2��y�n

2x� ��nx�ny� ��n2

x�

��

��

h1

h3

h4

h5

��

��

�

�2��x��y�ny� � ��2��2y�nx�ny�

��2��2y�nx�ny� � �2��x��y�n

2x�

��y�n2y� � ��x�nx�ny�

��y�nx�ny� � ��x�n2x�

��

�� 20�

Since h6 not occurs, this system gives only information about the shape of the ellipsebut not of its extension. Now, we superpose the two parts of the objective function un-der consideration of weighting factors. The weighting factors were chosen to be reci-procal to the variance of measured stresses and normal directions, respectively:

� � 1s2��1 �

1s2�n��2 � �21�


159

The approximated initial yield-locus curve is shown in Figure 8.10.It could be described well by a shifted von Mises yield ellipse. The initial values

of the deviatoric back stresses could be derived from the centre coordinates of the el-lipse to be �x0 � 8�3 MPa and �y0

� 3�2 MPa.

8.5 Bending of Notched Beams

8.5.1 Principle

Pure bending of a deeply notched specimen is a suitable test to investigate materialproperties also under conditions of different plane stress states. The geometry of thespecimen is shown in Figure 8.11.

The shape of specimen was optimized to get large ratios �y��x of the principalstresses in the material particles lying in the ligament. Bending tests have the advan-


160

Figure 8.10: Initial yield-locus curve: 0.01% offset-strain, half-axes ration a�b � ��3

and angle

� � 45 � given; �Fo � 218�3 MPa, �Mx � �19�8 MPa, �M

y � �14�7 MPa.

tages that with only one specimen, tension as well as compression can be analysed andthat a reversal loading can be realized easily in the experiment.

In this chapter, the determination of the strain distributions in the ligament of thespecimens for each load step is described.

8.5.2 Experimental technique

The in-plane Moire technique, about which was reported earlier [14], was used to mea-sure the deformation of the specimens. The geometric Moire is based on the superposi-tion of two gratings – the deformed object grating and the reference grating. EachMoire fringe, the so-called isothetic, describes the geometric location of all points withthe same Cartesian displacement component. This simple and clear fringe parameter isreflected in the following fundamental equation:

ux�x� y� � pmx�x� y� � �22�

where ux is the in-plane component of the displacement vector, mx fringe order, p pitch(Figure 8.12), x� y Eulerian coordinates. The preparation of the specimen (left) and amicrophotograph (right) of the object grating were presented in Figure 8.12.

Parallel to the Moire technique, strain gage measurements were carried out in thegrooves of notches and in the outer parts of the ligament.

As an example, the steps of loading (1–6), unloading (7) and reversal loading (8–14) of a notched bending specimen are stated in Table 8.1.

As typical Moire fringe fields in Figure 8.13, the isothetics are presented for theload step 4 from Table 8.1.

The pictures show high concentrations of the strain �x close to the grooves of thenotches and only small strains �y in the ligament. Because of that, an intense planestress state can be expected along the ligament.

Finally, Figure 8.14 demonstrates the remarkable effect that the deformations in thewhole specimen are removed almost completely at a defined reversal bending moment.


161

Figure 8.11: Specimen geometry for notched bending beams.


162

Figure 8.12: Moire technique.

Table 8.1: Load steps in bending of a notched specimen.

Loading Moment [Nm] Unloading Moment [Nm] Reversal loading Moment [Nm]

1 477 7 2 8 –932 552 9 –1873 609 10 –2734 655 11 –4305 689 12 –5036 736 13 –763

14 –817

Figure 8.13: Isothetic fields for load step 4 from Table 8.1 (left ux, right uy).

8.5.3 Approximation of displacement fields

By means of Moire measurements, we have got some measured values �ux��xs��ys� and�uy��xt��yt� of the displacements but at different points ��xs��ys� and ��xt��yt� for �ux and �uy,respectively.

To determine the values at given points like needed in our case, an approximationof the displacement fields will be done. The displacements ux and uy are independentfrom each other so that they could be handled separately. The used method is only de-scribed for ux, a detailed discourse is given in [15]. The general idea for approxima-tions is to use a local approach valid in a distinct area

�ux�x� y� ��m

am�m�x� y� �23�

with scalar factors am and properly chosen functions �m. Polynomial functions are wellestablished, but if one uses such with higher degree, the solution shows undesiredwave-like effects. On the other hand, it is useful to take functions, which are at leastC1-continuously to get also a good approximation of the strains.

The problem was solved by a Finite Element-like approximation. Good resultswere obtained using functions of the Serendipity-class of isoparametric 8-nodes-rectan-gular elements.

The mesh generation could be done with the preprocessor of an arbitrary Finite-Element program. If one takes the same mesh to approach both ux and uy, this proce-dure has the additional advantage that transformation of the mesh including the approx-imations into the initial configuration is possible. Hence, the description in material co-ordinates and the tracking of material points will be possible.

The functional to determine the scalar factors am was chosen to be:


163

Figure 8.14: ux-Isothetic field for load step 12 in Table 8.1.

� ��nei�1

1mi

�mi

j�1

��ux��xij��yij� � �uxij�2

� �� 1

�nei�1

Gi

�Gi

��u2x�xx � 2 �u2

x�xy � �u2x�yy�dGi

��

��

� �2

�nrr�1

lr

�lr

� �uxI�n � �uxII�n�2dlr

��

�� 24�

The first term consists of the least-squares sum between approximated and measureddisplacements of all ne elements. It will be weighted elementwise by the number mi ofmeasured points per element to exclude the influence of a possibly non-regular distribu-tion of measured points. The second term serves the smoothness of the approximationinside the elements. The third term reduces discontinuities in the first derivatives nor-mal to the boundaries of neighbouring elements.

In Figure 8.15, the approximated isothetics for the load step 4 are shown (cf. Fig-ure 8.13).

For bending specimens, small strains are coupled with finite displacements.Hence, the parameter identification demands the tracking of material points, the defor-mation fields were transformed into the initial configuration. The points in Figure 8.15are the originally measured positions of Moire fringes, whereas the isolines were nu-merically calculated from the Finite-Element approximation. One can observe a reallygood agreement between measured and approximated values.


164

Figure 8.15: Approximation of the isothetic fields from Figure 8.13.

In Figure 8.16, the deformations �x and �y in the ligament for load step 4 (cf. Fig-ures 8.13 and 8.15) are shown.

8.6 Identification of Material Parameters

8.6.1 Integration of the deformation law

The equations of elastic-plastic material behaviour, i.e. the deformation law (Equation(5)), the hardening rule (Equation (6)) and the flow condition (Equation (1)), could besummarized as:

E�1 �� F��

� �� 0 � �25�

�h� ��q��h� p� � 0 � �26�F��h� p� � 0 � �27�

The integration of these equations will be carried out for measured load steps �� ateach reference point of the ligament. Equation (25) represents ordinary differentialequations for the stresses �, Equation (26) such one for the internal variables h. Bothsystems are coupled and include the additional unknown scalar ��. The yield condition(Equation (27)) is needed as an additional algebraic condition.


165

Figure 8.16: Deformations �x and �y for load step 4.

Using implicit Euler time stepping, one finds the following system of non-linearequations:

E�1��n � �n�1� � ��n�F��n

�t � ��n � 0 � �28�

hn � hn�1 � ��nqn�t � 0 � �29�F��n�hn� � 0 � �30�

to compute the values �n, hn and �� for the n-th load step.To solve the non-linear Equations (28) to (30), a Newton method like in [16] was

used. With k as suffix for the number of iterates and � for the increments of the vari-ables, we get:

Jkn��h� ��

�

� �

��n � ��kn�F��

kn�t � E�1��kn � �n�1�

��knqkn�t � �hkn � hn�1��Fkn

��

��

�31�

with the matrix

Jkn �

E�1 � ��kn�2F��2

kn�t ��kn�2F��h

kn�t�F��

kn�t

� ��kn�q��

kn�t I � ��kn�q�h

kn�t qkn�t

�F��

kn�F�h

kn 0

��

��

� �32�

and the �k � 1�th iterate will be:

�k�1n

hk�1n��k�1n

�

� �

�kn

hkn��kn

�

��

��h� ��

�

� � �33�

To improve convergence for considerably large load steps ��n, it is possible to subdi-vide the load steps into a certain number of linear sub-load steps.


166

8.6.2 Objective function, sensitivity analysis and optimization

To determine the material parameters, the objective function

� � 12

�n

��1��Mn �p1� � ��M

n �2 � �2��Nn �p1��2 � �3��lK

n �p1� � ��lKn �2

� �4��uKn �p1� � ��uK

n �2� � minp

�34�

will be minimized. The calculated and measured values of bending stresses �Mn and

stresses �Nn caused by the normal force, and the longitudinal stresses �lK

n and �uKn at the

lower and upper notch grooves were compared. Moment M and normal force N werecomputed by numerical integration of stresses at the reference points along the liga-ment. To get the stresses ��M

n and ��Mn , the moment will be divided by the resistance mo-

ment W � 14bh2 for ideal plasticity, ��N

n is got by dividing the normal force by the

cross-sectional area. ��Nn will be zero for the four-point-bending specimen.

The usage of deterministic optimization algorithms requires the knowledge of the

gradientd�dp

of the objective function with sufficient accuracy. In our case caused by

the non-linear equations solved by implicit methods, the derivatives are not availableanalytically.

One way to compute the gradient is the numerical sensitivity analysis performedby the variation of the material parameters. For m parameters, the value of the objec-tive function has to be determined additionally m times for every iteration of thematerial parameters. The choice of parameter increments is unsure, and the control ofthe accuracy of the derivatives is difficult and also unsure.

Therefore, a semi-analytical sensitivity analysis will be preferred. The objectivefunction,

� � ��z�p�� 35�

depends indirectly on the parameters p. The total differential is:

d� � ��z

� �T�z�p

dp � aTdp � �36�

a is called the sensitivity vector and gives a measure for the sensitivty of the objectivefunction at arbitrarily parameter changes [17].

The derivatives��z

could be determined easily from the objective function. The

only difficulty is the determination of�z�p

. This will be done by implicit differentiation

similar to [15, 18].The system G of Equations (28) to (30) depends on the variables �, h and ��.

These will be combined to the enlarged state vector:


167

z�p� � ��p��h�p�� T � �37�

which yet depends on the vector of material parameters. The iterated system (Equations(28) to (30)) to compute the n-th load step could be summarized as:

Gn�zn�p�� zn�1�p�� p� � 0 � �38�

Implicit differentiation gives:

dGn

dp� �Gn

�zn

�zn�p

� �Gn

�zn�1

�zn�1

�p� �Gn

�p� 0 �39�

so that

�Gn

�zn

�zn�p

� � �Gn

�zn�1

�zn�1

�p� �Gn

�p�40�

is a system of linear equations to determine�zn�p

. The matrix�Gn

�znrepresents the iterated

Jacobean Jn from Equation (32), which is known in its decomposed form.Furthermore is:

�Gn

�zn�1�

�E�1 0 00 �I 00 0 0

�

� �41�

and

�Gn

�p�

��n�2F��p

� ��n�q�p

�F�p

��

��

�42�

have to be supplied by the material subroutine. Under consideration that�z0

�p� 0, it is

possible to compute the derivatives fo the next load step from that of the previous one

with the accuracy of the integration of the deformation law.To solve the optimization problem of parameter identification (Equation (34)),

several gradient-based optimization methods like steepest descent, BFGS, Gauß-New-ton and Levenberg-Marquardt’s methods were tested. The Levenberg-Marquardt algo-rithm, which combines steepest descent and Gauß-Newton method [19, 20], was pre-ferred as a very robust and suitable procedure for the non-linear least-squares optimiza-tion. Hereby, the parameter vector will be changed according to


168

pw�1 � pw � sw �43�

with the search direction

sw � � �wGN � I

� ��1��w � �44�

where �wGN means the actual Gauß-Newton matrix.

Far from the solution, will be taken large so that the procedure is nearly a stee-pest descent, and in the neighbourhood of the solution, will be taken small so that itis nearly a Gauß-Newton algorithm. The control of the choice of the parameter willbe done in such a manner that the next iterate will be searched in a “model-trust re-gion” with �sw� � .

8.6.3 Results of parameter identification

The implemented generalized material model consists of the following equations:

• quadratic yield function (Baltov and Sawczuk [21])

F � Nijkl

��Dij � �ij

��Dkl � �kl

�� 2

F��plv � � 0 � �45�

• evolutional relations• yield stress – isotropic hardening

�F � A��plv � � �46�

• kinematic hardening by Backhaus [22]

��ij � B��plv ��pl

ij � �47�

• distorsional hardening by Danilov [23]

�Nijkl � C��plv �

��plv

��plij ��

plkl � �48�

where A, B and C could be chosen as arbitrary functions of the equivalent plastic strain�pl

v .Some special formulations will be at disposal:• yield function (shifted von Mises)

F � 32��D

ij � �ij��Dij � �ij� � �2

F � 0 � �49�


169

• yield stress– modified power law

�F � �Fo � a1��plv � a2�a3 � aa3

2 � � �50�

– arctan law

�F � �Fo � a1 arctan �a2�plv � � a3�

plv � �51�

• kinematic hardening– by Prager (cf. [22])

��ij � b1 ��plij � �52�

– by Armstrong-Frederick (cf. [24])

��ij � b1 ��plij � b2 ��

plv �ij � �53�

All computations have been carried out with the initial values for �Fo, �11o and �22o

taken from the approximation of the initial yield-locus curve (cf. Figure 8.10).The power law (Equation (50)) gave no satisfying results because the parameters

depend on each other and it shows no asymptotic behaviour of the uniaxial flowcurves.

Therefore, the arctan law (Equation (51)) for isotropic hardening combined withkinematic hardening and initial anisotropy has been used for further computations. Re-sults are shown in Figures 8.17 and 8.18.

The consideration of kinematic hardening by Armstrong-Frederick [24] and ofdistortional hardening by Danilov [23] did not reveal any quantitatively better approxi-mation.

8.7 Conclusions

In the paper, some possibilities of bending tests to identify material parameters of in-elastic materials were described. For that reason, the co-operation between theoreticallyand experimentally working academics has to be much closer, as this is usual at ger-man technical universities. The foundation of the Deutsche Forschungsgemeinschaft –Collaborative Research Centre (Sonderforschungsbereich 319) – has been proved as avery effective measure to overcome reservations and to realize such a close co-opera-tion.

An important result of this project is that bending tests are suitable very well tostudy elastic-plastic material properties. Bending tests have the fundamental advantages


170

that in one experiment, tension as well as compression exist and that a reversal loadingto identify kinematic strain hardening can easily be realized. In the experiments, somedifficulties result from the fact that strains in a very large extension must be detected.So, for definition of yield stresses, the offset strain is about 10–4, and on the other side,a maximum strain of 5 ·10–2 should be measured. In both cases, the stress level is ap-proximately the same. This taking into account, stable numerical results for the elasticconstants, the yield stresses as well as the yield curves and the anisotropic yield-locuscurve were obtained from bending tests of specimens.

The identification of hardening parameters was carried out by analysing the dis-placement fields of notched specimens. The suitability of such experiments has beenproved. Further investigations should use all informations about the whole experimen-tally investigated area. Therefore, the use of the Finite-Element method to calculate thenumerical comparative solution is absolutely necessary.

Shortened calculations as described here require low computational times andmay give a deep insight into the effects of several material models in combination with

8.7 Conclusions

171

Figure 8.17: Parameter identification for the bending test of the notched specimen, arctan law forisotropic hardening, kinematic hardening by Prager (cf. [22]), �Fo � 218�3 MPa, �11o ��8�3 MPa, �22o � �3�2 MPa; optimized parameters: a1 � 94, a2 � 1626, a3 � 380, b1 � 1600;mean quadratic deviation: 20.7 MPa.

a special analysis of measured data. Such calculations could be used also furthermoreto get reliable starting values for the Finite-Element-based parameter identification.

The authors’ hope that they can continue the research about the relatively newidea to identify inelastic material properties and parameters by means of the evaluationof inhomogeneous strain and stress fields. For instance some possibilities are given inthe DFG project [6].

Acknowledgements

The authors would like to thank Prof. Dr. Dr. E.h.E. Steck and Prof. Dr. R. Ritter fromTechnical University of Braunschweig for their extensive support during the prepara-tion and realization of this project.


172

Figure 8.18: Related uniaxial flow curves to Figure 8.17.

References

[1] E. Stein, D. Bischoff, R. Mahnken: Identifikation mit Finite-Element Methoden. Arbeits-und Ergebnisbericht 1991–1993. Subproject B8, Collaborative Research Centre (SFB 319).

[2] E. Stein: Parameteridentifikation mit Finite-Element Methoden. Forderungsantrag 1994–1996. Subproject B8, Collaborative Research Centre (SFB 319).

[3] R. Mahnken, E. Stein: Parameter Identification for Inelastic Constitutive Equations Basedon Uniform and Non-Uniform Stress and Strain Distributions. This book (Chapter 12).

[4] K. Andresen, S. Dannemeyer, H. Friebe, R. Mahnken, R. Ritter, E. Stein: Parameteridentifi-kation fur ein plastisches Stoffgesetz mit FE-Methoden und Rasterverfahren. Bauingenieur71 (1996) 21–31.

[5] R. Kreissig, J. Naumann: Weiterentwicklung der Theorie der plastischen Verfestigung undihre experimentelle Verifikation mit Hilfe des Moireverfahrens. DFG-Projekt 1994–1996/Zwischenbericht 1996.

[6] R. Kreissig, A. Meyer: Effiziente parallele Algorithmen zur Simulation des Deformations-verhaltens von Bauteilen aus elastisch-plastischen Materialien. Forderungsantrag 1996–1998. Subproject D1, Collaborative Research Centre (SFB 393).

[7] R. Kreissig: Parameteridentifikation inelastischer Deformationsgesetze. Technische Mecha-nik 16(1) (1996) 97–106.

[8] H. Herbert: Über den Zusammenhang der Biegungselastizitat des Gußeisens mit seinerZug- und Druckelastizitat. Mitt. und Forschungsarbeit. VDI 89 (1910) 39–81.

[9] A. Nadai: Plasticity. McGraw-Hill, New York, London, 1931.[10] V. Laws: Derivation of the tensile stress-strain curve from a bending data. J. Materials Sci.

16 (1981) 1299–1304.[11] R. A. Mayville, I. Finnie: Uniaxial stress-strain curves from a bending test. Exp. Mech.

22(6) (1982) 197–201.[12] M. Stockmann, J. Naumann, P. Bormann, F. Pelz: Zur Widerstandsanderung von Dehnungs-

meßstreifen bei großen Deformationen. Materialprufung 38(4) (1996) 134–138, 38(5)(1996) 216–219.

[13] P. Bormann, J. Naumann, M. Stockmann: Biegeversuche zur Ermittlung einachsigerFließkurven fur Zug und Druck. In: O.T. Bruhns (Ed.): Große plastische Formanderungen.Bad Honnef, 1994. Mitteilungen des Inst. fur Mechanik, Nr. 93, Ruhr-Universitat Bochum.

[14] J. Naumann: Grundlagen und Anwendung des In-plane-Moireverfahrens in der experimen-tellen Festkorpermechanik. VDI-Fortschrittsberichte, Reihe 18, Nr. 110, Dusseldorf, 1992.

[15] E. Bohnsack: Continuous field approximation of experimentally given data by finite ele-ments. Computer & Structures 63(6) (1997) 1195–1204.

[16] D. Michael, A. Meyer: Some remarks on the simulation of elasto-plastic problems on paral-lel computers. Preprint-Reihe Chemnitzer DFG-Forschungsgruppe “Scientific Parallel Com-puting”, Technische Universitat Chemnitz-Zwickau, March 1995.

[17] H. Eschenauer, W. Schnell: Elastizitatstheorie. BI-Wissenschaftsverlag, Mannheim,1993.[18] R. Mahncken, E. Stein: Identification of Parameters for Visco-plastic Models via Finite-Ele-

ment Methods and Gradient Methods. IBNM-Bericht 93/5, Institut fur Baumechanik undNumerische Mechanik der Universitat Hannover, 1995.

[19] J.E. Dennis, R. B. Schnabel: Numerical Methods for Unconstrained Optimization and Non-linear Equations. Prentice Hall, Englewood Cliffs, 1983.

[20] S.S. Rao: Engineering Optimization. Wiley & Sons, New York, 1996.[21] A. Baltov, A. Sawczuk: A Rule of Anisotropic Hardening. Acta Mechanica 1 (1965) 81–92.[22] G. Backhaus: Deformationsgesetze. Akademie-Verlag, Berlin, 1983.[23] V. L. Danilov: K formulirovke zakona deformacionnogo uprocnenija. Mechanika tverdogo

tella, Moskva 6 (1971) 146–150.[24] V. Dorsch: Zur Anwendung und Numerik elastisch-plastischer Stoffgesetze. In: Pro-

zeßsimulation in der Umformtechnik, Band 9, Springer, Berlin Heidelberg, 1996.

References

173

10 On the Behaviour of Mild Steel Fe 510under 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 materialbehaviour of mild steel Fe 510 under multiaxial cyclic plastic loading.

First of all, detailed information about the material response under cyclic plasticloading are necessary. Therefore, extensive experimental investigations are made in-cluding uniaxial single- and multiple-step tests and biaxial tension-torsion tests withvarious prestrains, increasing or decreasing strain amplitudes, and several proportionaland non-proportional biaxial loading paths, respectively. Cyclic hardening or softeningin the uniaxial case, or additional hardening under non-proportional loading are someof the observed effects.

To describe the material peculiarities, the two-surface model of Dafalias-Popovhas been modified. The improvements result in a rate-independent, isothermal two-sur-face model, which is presented here.

Experimental data from the uniaxial and biaxial tests, and, in addition, from testson structural components are compared with corresponding calculations made with thenew model.

The results demonstrate the capabilities of the extended-two-surface model to pre-dict the behaviour of mild steel and steel constructions under multiaxial cyclic plasticloading.

218

* Technische Universitat Braunschweig, Institut fur Stahlbau, Beethovenstraße 51,D-38106 Braunschweig, Germany



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 charac-teristics of the different heats used in the investigations can be found in the correspond-ing papers (Scheibe [1], Reininghaus [2]).

Two types of specimens were used in the investigations. Figures 10.1 and 10.2show sketches of both the cylindrical specimens used in the experiments with uniaxialloading, and the tubular specimens for the biaxial investigations.

Different servohydraulic testingmachines wereused in the uniaxial investigations.Thebiaxial investigations were performed on a 160 kN-tension-compression “SCHENK” uni-versal 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 3.5and 24‰/min. For the force-controlled experiments, the loading rate was 3 kN/s. All


219

Figure 10.1: Cylindrical specimen for uniaxial experiments.

Figure 10.2: Tubular-type specimen for biaxial experiments.

tests were performed at room temperature. The varied parameters in the uniaxial loadedtests 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 se-quence of loadings, the evolution of cyclic hardening and softening, the relaxation ofmean stresses, and the size of the elastic region.

To investigate the ratchetting effects, stress-controlled experiments were per-formed varying the initial strain �m, the stress amplitude �a, the mean stress �m, andthe stress ratio R��u/�o.

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 Fig-ures 10.4 and 10.5.

In these figures, the courses of the half-range of stresses vs. the number of half-cycles are shown. In addition, the strain vs. time is plotted for explanation in the upperleft part of the diagram. The resulting stress vs. strain is given in the upper right part.

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

220

Figure 10.3: Varied parameters of the uniaxial experiments.

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

After passing the linear-elastic region, the distinct yield point and the yield pla-teau, mild steel Fe 510 shows the well known Bauschinger effect if the specimen is un-loaded. During the cyclic loading, the maximum stresses of a hysteresis-loop are notconstant: For amplitudes �a smaller than 5‰, the maximum stresses decrease from cy-cle to cycle, and a saturated state is reached after 500 or more cycles (cyclic softening).Amplitudes �a higher than 5‰ cause an increase of maximum stresses during the first40 cycles (cyclic hardening).

Note that cyclic softening or hardening is understood as a decrease or an increase ofthe stress amplitude in comparison with the stress level of the monotonic stress-straincurve at this strain level. For �a�12‰, the stress level of the monotonic stress-straincurve is almost constant (with ��F). Therefore, cyclic hardening or softening can beobserved easily. Stress levels ��/2 (for symmetric amplitudes) greater than �F are seenby definition as cyclic hardening, and stress levels ��/2 lower than �F as cyclic softening.

A cyclic stress-strain curve is achieved by plotting the corresponding stress ranges��/2 at the stabilized states vs. the corresponding strain amplitudes �a. Figure 10.6shows 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 cyclicsoftening and cyclic hardening.


221

Figure 10.5: Multiple-step test (RFE), �a �2-12-2‰.

Figure 10.6: Cyclic stress-strain curves.

The cyclic stress-strain curves obtained after 40 cycles and the one after 500 cyclesdiffer if smaller amplitudes are used. This shows that a saturated state is not reached after 40cycles at all. With higher amplitudes, the two curves tend to become identical: The steadystate is reached during the first 40 cycles. However, after prestraining with a higher am-plitude (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 �m�+40‰.The maximum stress at 52‰ (or the maximum stress level ��/2) decreases within thefirst cycles. Note that this transient behaviour is not a cyclic softening according to theabove definition because, during the first loops of the amplitude �a�12‰, only themean stress caused by the mean strain returns to zero.

Parallel to this so-called mean-stress relaxation, a cyclic hardening of the�a�12‰ amplitude takes place, but due to the dominance of the mean-stress relaxa-tion, only a softening of the stress level ��/2 can be observed.

A comparison of the stabilized loops for amplitudes �a�12‰ with different meanstrains shows that the different mean strains do not significantly influence the shape orthe maximal stress amplitudes of the hysteresis-loops.

Figure 10.8 shows the cyclic stress-strain curves for different mean strains. It isseen that the maximum stress amplitude ��/2 depends only on the strain amplitude �a.This result is the basis for the above definition of cyclic hardening or cyclic softeningunder a given strain amplitude.

Another important point besides the effect of cyclic hardening is the influence ofthe cyclic loading on the size of the elastic region. The elastic regions were determinedusing an offset-proof-strain method (see Scheibe [1] for further explanation) with aproof 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 cyclicloading. The diagram shows that the size of the elastic domain k is a function of themaximum strain amplitude �max. If the former is greater than 5‰, the size of the elasticregion is reduced to the value ks�100 MPa. For strain amplitudes lower than 5‰, theelastic region decreases very slowly (�a�2‰ in Figure 10.9). The dashed line of theamplitude �a�2‰ shows that the size of the elastic region tends to reach the same val-ue as that of the greater amplitudes within some hundred cycles.


222

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

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) quick saturation, especially with small �a and �m,b) positive increase of strain without saturation,c) negative increase of strain with gradual saturation (at a large number of cycles),d) 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 thekind of stress (initial cross-section: technical stress, and actual cross-section: effectivestress) 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 constantmaximum stress (Figure 10.11), and, in addition, a distinct mean stress (34.5 MPa), whereno ratchetting is found (Figure 10.12). The amount of the occurred ratchetting depends onthe difference between the actual mean stress and the mean stress, which leads to zeroratchetting (Kuck [3]).


223

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

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


224

Figure 10.10: Different appearances of the ratchetting effect.

Figure 10.11: Influence of R � �u��o�

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 loadingpaths, the loading intensity �B (defined by Equation (1)), and the ratio � between ten-sile and torsional loads (defined by Equation (2)) were varied. Note that �B is not a me-chanical derived equation, it is used only to compare biaxial and uniaxial loads here.For the experimental investigations, different loading intensities �B between 1.90‰ and7.10‰ were chosen.

�B ��2

a �13�2

a

�� 1�

� ��3

� �a

�a� �2�

The achieved stress curves and the calculated uniaxial equivalent stresses (based on thetheory of v. Mises) show the influences of various sequences of loading paths or theadditional-hardening effect.

Moreover, the yield-surface investigations give additional insight into the evolu-tion of the elastic region under complex loadings. Here, the influence of the intensity�B and the loading path on the location, shape, and size of the measured yield surfacewere investigated.

Additional experiments were made to determine the influence of several param-eters coming out of the process of yield point probing itself, i.e. technique and se-quence of yield point probing, choice of the starting point, or fixing of the elastic mod-ulus.


225

Figure 10.12: Influence of the mean stress.

10.2.3.2 Relations of tensile and torsional stresses

Figure 10.14 gives some characteristic results of the evolution of the tensile and tor-sional stress in the transient state of biaxial cyclic-loaded mild steel.

The evolution of the stresses of the biaxial proportional and non-proportional testsdiffers with increasing intensities. For a proportional loading, only the maximum stress-es increase and the stress curve grows without changing its shape (Figure 10.14 a).


226

Figure 10.13: Biaxial loading paths.

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

�


227

a) Path 03; � � 1�0 b) Path 07; � � 1�0

c) Path 10; � � 1�0 d) Path 09; � � 1�0

e) Path 09; � � 1�0 f) Path 09; � � 2�4

g) Path 08; � � 1�0 h) Path 08; � � 1�0

On the other hand, for some of the non-proportional loadings, a significant change inthe shape of the stress course in addition to the increase of the maximum stresses can befound (Figure 10.14 b,g–i). Note that the loading path and the relation � between �a and �a

does not change. The results of a varying � can be seen in Figure 10.14 d–f. In Fig-ure 10.15, the uniaxial equivalent stresses �v are plotted vs. the maximum equivalentstrain or intensity �B. In these experiments, the cyclic hardening in the uniaxial cyclicstress-strain curve starts at about 3‰. The difference between the cyclic stress-straincurve of the uniaxial experiments described in Section 10.2, and the curve seen in Fig-ure 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 pro-portional loadings, no significant difference in the maximum stresses between the uni-axial and the biaxial proportional loadings can be found. A significant difference be-tween the cyclic stress-strain curve and the equivalent stresses is found however fornon-proportional loading paths. This effect is named additional hardening here.

The additional hardening appears to be strongly dependent on the type of non-proportional loading. In general, the additional hardening increases by an increasingphase angle � (Figure 10.13). For higehr values of � (between 60 and 90 degrees, path06 and 07), the additional hardening is almost constant. The amount of additional hard-ening of path 10 corresponds to that of path 07. The highest amount of additional hard-


228

i) Path 08; � � 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-proportionalloadings.

ening is found in experiments with the strain path 09 (“Butterfly”). Here, the equiva-lent stress in the saturated state reaches the level of the uniaxial tensile strength. Theadditional 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 firstuniaxially loaded (�a�6‰) up to a saturated state, then underwent a non-proportionalloading path 07 (�B of 8‰), and finally was uniaxially loaded again.

The cyclic hardening during the first loading stage is followed by an additionalhardening in the second stage. During the third loading, the additional hardening fadesso that the saturated stress-strain curves of the first and the third loading level arenearly identical.

10.2.3.3 Yield-surface investigations

Yield surfaces were investigated in the transient and in the saturated state of thematerial behaviour. To reduce time-consumptional manual controlling, a computer pro-gram was developed, which allows yield surfaces at any point of any complex loadingsto be established automatically. This program and the experimental set-up is describedcomprehensively by Dannemeyer [4].

An offset-proof-strain method was used to define the onset of plastic flow. For anincreasing offset-proof strain, the yield surfaces of mild steel Fe 510 in saturation showthe same effects as described for other materials (e.g. Michno and Findley [5]). The distinctcorner in preloading direction and the flattening opposite to it tend to blur, and the size ofthe surface increases, mainly against the preloading direction (Figure 10.16). TheBauschinger 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 inthe saturated state. To determine yield surfaces in the transient state of the material be-haviour, the offset-proof strain was reduced to 0.03‰. A further decrease of the offset-proof strain was restricted by the precision of the technical set-up.

In this investigation, several yield-surface determinations were performed on asingle specimen. 16 yield probings with different ratios of sizes of tensile and torsionalstrain increments are combined to build up a yield surface. To investigate the influence


229

Figure 10.15: Uniaxial equivalent stresses of biaxial loading paths.

of the sequence of the different yield-probing directions, yield surfaces were deter-mined using two different sequences (Figure 10.17).

For materials, which reach a cyclically stable state during cyclic loading, the in-fluence of the probing sequence can be excluded if the specimen is loaded with severalcycles between two yield probings. The small hardening effects drawn by the previousyield probing disappear completely, and a similar loading state can be reached at thebeginning of every new yield probing (Figure 10.18). This allows yield surfaces to beinvestigated with negligible dependence on the probing path (Figure 10.19). (The turn-over points and the first yield points are marked with arrows here. The loading path foreach of the yield surfaces is given by symbols.)

This method is called the single-point technique (SPT) here, in contrast to the mul-tiple-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.


230

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

Figure 10.17: Sequences of yield point probings.

��


231

Figure 10.18: Single-point technique.

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

��

The influence of the probing sequence on a yield surface determined with themultiple-probing technique is systematic by nature (Figure 10.20). So, the measuredyield surface can be corrected, at least in quality, if a surface investigated with the sin-gle-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 loadingand the elastic region after the turnover of the 20th cycle of a specimen under uniaxialtension-compression load is plotted.

In order to compare the two gradients, the turnover point of the 20th cycle ismoved to the origin. It is seen that the slope of the initial loading curve is nearly con-stant over the plotted range of 1.3‰. After being loaded 20 times into the plasticrange, a distinct proportional region is missing.

This non-linearity of the stress-strain curve under preloading affects the yield-sur-face 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, ifpossible in the centre of the elastic region to secure an almost rectangular touch of theelastic-plastic border.

Due to the non-linear area after a turnover point, the unloading strain �s becomesa parameter of the size of the established yield surface.

The results of an experiment, shown in Figure 10.22, demonstrate the effect ofvarious unloading strains on the yield surface of a uniaxial cyclic tension-compression-loaded specimen. It is obviously that the expansion of the yield surface against the pre-loading direction depends on the amount of the unloading strain. The diameter of theyield surface rectangular to the preloading direction is not affected, however.

After preloading, the lack of a distinct area of proportionality influences the yield-surface determination not only in the setting of a starting point but also in the calculationof plastic strains. Using an offset-proof-strain definition in combination with an automa-


232

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

�� 3��

tized experimental procedure, a continuous separation of elastic and plastic strains has to becarried out on-line during a yield probing. The elastic moduli (E and G) can be set either toconstant values for the whole experiment, or they can be determined from the gradient of adefined number of data considered to be linear-elastic. If the current elastic moduli changeduring a cyclic loading and constant moduli are considered in the on-line calculation ofplastic 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 cur-rent elastic characteristics. In comparison to the Figures 10.19 and 10.20, it can be seenthat the size of the yield surface depends strongly on how the elastic moduli are given.


233

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

Figure 10.22: The effect of various unloading strains �s on yield loci under cyclic uniaxial ten-sion-compression loading.

��

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. Ifnon-proportional loading paths are used, the difference in size of the SPT- and MPT-loci is at its greatest (Figure 10.23, upper left yield surface).

In the transient state of material behaviour, yield-surface investigations are moresensitive than in the saturated state. Even in the area of the yield plateau, a yield-pointdetermination with its incursion into the plastic region influences strongly the lateryield-point probes. In general, the intensive hardening during the first few cycles ex-cludes 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 small-est 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 movein the stress space immediately after the elastic limit is passed. In addition, a distortion ofthe initial round yield surface (�-

��3

��-space) occurs. The pronounced isotropic softening

is strongest during the first cycle, and in the case of a proportional load, it is usually com-pleted 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 almostcompleted at the beginning of the second cycle (Figure 10.24). This isotropic softening isalways connected with a kinematic hardening. In further states of the cyclic loading, theyield 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 stress-es �v, and from the second to the fifth cycle, a distinct increase of the maximum stress-es obtained by a movement of the yield surface takes place. It can be stated that the ad-ditional hardening of a non-proportional load is also a type of kinematic hardening.


234

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

��

This distortion of the yield surface (formative hardening), including the deviationof the initial round shape by forming a corner in preloading direction and a flat side atthe opposite, is the third mechanism of hardening besides the isotropic and kinematichardening 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 preloadingdirection and a flattening opposite to it. In addition, the two diameters of the yield sur-face change differently during a proportional load. In the saturated state, the diameterin 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 onlyto 70% as seen in Figure 10.19, for example. A non-proportional load forms a more


235

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

Figure 10.25: Definition of diameters of a yield surface.

��

rounded and smoother shape, and the diameters decrease more homogeneously becauseof the changes in the direction of the load increments.

An approximate alignment of the yield surface with the corner and the flatteningtowards the direction of the stress increment can be found only at proportional loadingpaths. At some of the non-proportional loading paths, the direction of the stress incre-ment at the turnover point differs distinctly from the direction of the suggested axis ofsymmetry 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 describethe 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 thetwo-surface model of Dafalias and Popov [8] represents a suitable basis for further de-velopments. The extensions of the original two-surface model are presented here. Thenew 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], andReininghaus [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 cal-culations shown in this paper were carried out with the same set of parameters. Therewas no extra fitting necessary for any special kind of loading path or calculation.


236

10.3.1.2 Loading and bounding surface

The original two-surface model of Dafalias and Popov assumes that the yield surfaceor loading surface and the memory or bounding surface (Figure 10.26) harden kinemati-cally and isotropically, while the two surfaces are in contact. If there is no contact be-tween the two surfaces, the memory surface remains unchanged, while the loading sur-face moves according to the Mroz rule [13]. This rule secures a tangential contact with-out any intersection if the two surfaces come into contact.

Both the loading surface and the bounding surface are assumed as hyperspheresin the deviatoric stress space and are represented as:

F � 12��D �

��

��D �

�� 1

3k2 � 0 � �3�

�F � 12��D � �

��

��D � �

�� 1

3�k2 � 0 �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,

��D 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 �in (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-


237

Figure 10.26: Loading and bounding surface.

nal model of Dafalias and Popov, the bounding surface in the ETS-model is only ableto harden or soften isotropically. Furthermore, two bounding surfaces instead of one areimplemented. The inner one corresponds to the original bounding surface in the modelof Dafalias and Popov. The outer one is used as a control surface and is activated onlyunder non-proportional loadings.

The plastic modulus in the ETS-model is calculated from Equation (5):

P � �P0 � h�

�in � ��5�

with

� distance between the actual stress point and the bounding surface (see Figure 10.26),�in distance between the yield point and the bounding surface (see Figure 10.26),�P0 plastic modulus of the strain hardening region,h shape parameter.

The plastic modulus P is split up into a modulus for the kinematic hardening P� andthe isotropic hardening Pk:

P � P� � Pk � �6�

Pk is calculated from the formulation of the isotropic hardening or softening of theyield surface. Pk is a function of the size of the new implemented memory surface inthe strain space. The plastic modulus P� for the kinematic hardening is determinedfrom the difference between the total plastic modulus P and the plastic modulus for theisotropic hardening Pk.

During the development of the model, different rules for the kinematic hardeningof the surface were tested (Reininghaus [2]). The best results are obtained with theoriginal kinematic hardening rule of Mroz [13], therefore all results presented here arecalculated with this rule.

10.3.1.3 Strain-memory surfaces

The strain-memory surfaces in the strain space allow the monotonic and cyclic materialbehaviour 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 strainamplitude during the whole cyclic loading:

Mm � 12��p �

��m��

��p �

��m� � 3

4q2

m � 0 � �7�


238

qm ��

dqm � �8�

dqm � 12Hm

�n�nm�d�vp � �9�

��m �

�d��m � �10�

d��m � 1

2Hm d

��p �11�

with

Hm � 0 forMm � 0Mm � 0 and

�n�nm� � 0 �

��12�

Hm � 1 for Mm � 0 and�n�nm� 0 � �13�

�n normal to the loading surface,

�nm� normal to the monotonic strain-memory surface,

��p plastic strain tensor,

��m centre of the monotonic strain-memory surface,d�vp equivalent plastic strain increment,qm 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 anadditional factor (cs in Equations (15) and (16)). The change in the size of this memorysurface describes cyclic hardening or softening:

Ms � 12��p �

��s��

��p �

��s� � 3

4q2

s � 0 � �14�

dqs � �1 � Hm�Hscs�n�ns�d�vp � �15�

d��s � �1 � Hm�Hs�1 � cs� � Hm�d

��p � �16�

qs ��

dqs � �17�

��s �

�d��s �18�


239

with

Hs � 0 forMs � 0Ms � 0 and

�n�ns� � 0 �

��19�

Hs � 1 for Ms � 0 and�n�ns� 0 � �20�

�ns� normal to the saturated strain-memory surface,

��s centre of the saturated strain-memory surface,qs size of the saturated strain-memory surface,cs factor for the isotropic hardening per cycle.

• Strain memory Ma for the actual loading direction

The memory surface Ma for the actual loading direction will decrease to zero when theangle between the normal vector of the loading surface and the normal vector of thememory surface Ma exceed 90�:

Ma � 12��p �

��a��

��p �

��a� � 3

4q2

a � 0 � �21�

dqa � 12 �n�na�d�vp � �22�

d��a � 1

2d��p � �23�

qa ��

dqa � �24�

��a � �1 � Ha�

��p �

�d��a �25�

with

Ha �1 for

�n�na� � 0

0 for�n�na� 0

�� 26�

�na� normal to the actual strain-memory surface,

��a centre of the actual strain-memory surface,qa size of the actual strain-memory surface.


240

10.3.1.4 Internal variables for the description of non-proportional loading

The variable Z is used to distinguish between a proportional and a non-proportionalloading within the current loading increment. Z is defined as:

Z � 1 � d D

�d��D�

D

��D� for �d

��D� � 0 � �27�

Z � 0 for �d��D� � 0 � �28�

The internal variables FS and FL are functions of Z and d�vp. In the case of propor-tional loading, FS and FL are assigned to zero. During non-proportional loading, bothvariables rise to the value one. During the process, FL has a temporal delay to FS. Ef-fects, which occur immediately with the set-in of a non-proportional loading, are con-trolled by the variable FS, and those processes, which occur slowly during a non-pro-portional loading, are controlled by FL. If the non-proportional loading is followed by aproportional one, both internal variables decrease to zero again to simulate the erasureof the additional hardening found in the experiments (see Section 10.2.3.2):

FS ��

dFS � �29�

dFS � W2�Z � FS�d�vp �30�

with W2 � 0�1 tanh�qm�q�� for Z FS �W2 � 0�01 for Z � FS �

�

FL ��

dFL � �31�

dFL � W3�Z � �1 � cos 30��d�vp �32�

with W3 � �1 � FL�0�1 for Z 1 � cos 30� �W3 � FL for Z � 1 � cos 30� �

�

��

��


241

10.3.1.5 Size of the yield surface under uniaxial cyclic plastic loding

The size of the yield surface k is defined to be:

k � ks � �k0 � km�2�10qmq� � �ks � km�2�10qs

q� �33�

withkm � 0�6 k0 size of the yield surface after the first plastic loading,ks � 0�4 k0 size of the yield surface for the saturated state.

10.3.1.6 Size of the bounding surface under uniaxial cyclic plastic loading

The mathematical formulation of the bounding surface has to describe a curve, whichis formed from the parts of the monotonic or the cyclic stress-strain curve, whichdelimit the current maximum stress for a given strain (see Figure 10.15).

• Section I for qm � q� � qs � q� ��k � const � k0 � �34�

• Section II for qm q� � qs � q� ��k � k0 � 0�73 �qm � q�� 0�0162 �qm � q��2 � �35�

• Section III for qm q� � qs q��

�k � k0 � 0�73 �qm � q�� 0�0162 �qm � q��2 � ��k 1 � qs

q�� 5

� qs � q�qm � q� �36�

with

k0 initial size of the yield surface,��k cyclic hardening due to uniaxial loading.

10.3.1.7 Overshooting

The overshooting problem is connected with the update of the initial value �in. To pre-vent the overshooting effect, the initial value �in is limited to:

�in�min � 0�35 �max with �max ��23

�2��k � k� � �37�


242

10.3.1.8 Additional update of �in in the case of biaxial loading

In the case of uniaxial loading, a new �in has to be determined as soon as there is anangle higher than 90 � between the normal vector of the strain-memory surface Ma andthe normal vector of the yield surface. In the case of biaxial loading, an additional up-date is made if the angle between the normal vector of the yield surface and the currentdeviatoric stress vector exceeds 30 �.

10.3.1.9 Memory surface F

The memory surface F is defined as a surface in the stress space, which allows themodel to remember a previous or current non-proportional loading. The surface is con-nected with the internal variables FS and FL. Its formulation is similar to the v. Misesyield rule:

F � ��

��

��

�� 2

3k 2 � 0 � �38�

In the case of uniaxial loading, this surface corresponds to the loading surface in sizeand location. During a non-proportional loading, the memory surface shows an addi-tional isotropic hardening:

k � 2�0�

P d�vp � k �39�

with

P � 5�0 �k max � k �FS � �k � k��1 � FS�2 ��

�in

� �2

� �40�

k max � k PARFS � k � �41�

P plastic modulus for the isotropic hardening of the memory surface,k actual size of the memory surface,k max maximal size of F ,k PAR maximal difference between the size of the loading surface and this memory sur-

face.

If the current stress point lies within the memory surface, the location of F remains un-changed. For a contact of the stress point with the memory surface, the surface movesin a way that the normal vector of the memory surface corresponds to the normal vec-tor to the yield surface. Meanwhile, the stress point remains on the memory surface.The displacement of this surface is defined as:


243

��

��D � �

��D �

�� k

k� �42�

An additional considerable influence on the predicted material behaviour is obtained bya modification of the update behaviour of �in, in combination with the definition of F .For a stress point lying within the memory surface, the value �in is constantly updatedso that the material behaves in a quasi-elastic manner. However, in regions with�in � �in�min, the constant value �in�min (see Equation (37)) instead of the updated �in istaken into consideration.

10.3.1.10 Additional isotropic deformation of the loading surfacedue to non-proportional loading

The isotropic deformation of the memory surface F and the loading surface F differsby the amount of 2�0

P d�vp (see Equation (39)). Parallel to the isotropic deformation

of the memory surface F , the loading surface changes in a way that a hardening of thememory surface causes an additional softening of the loading surface. The amount ofthe change of the loading surface is

P d�vp.

The incremental deformation of the loading surface is given by:

dk � �k0 � km� bq� a

bqmq� ln a

� �dqm � �ks � km� b

q� abqsq� ln a

� �dqs

� ��

P d�vp �

�43�

The additional deformation of the loading surface can only be expressed by an incrementalform. The incremental equations are formulated in a way that the integrated incrementsconverge to limits. These limits are the known maximal values of the parameters.

During a cyclic multiaxial loading, where the stress point is located mostly on theloading surface, this additional deformation of the loading surface generates a deforma-tion of the calculated stress path.

10.3.1.11 Additional isotropic deformation of the bounding surfacedue to non-proportional loading

To describe effects of biaxial loading, an additional isotropic deformation of the boundingsurface is necessary. Therefore, the additional parameters FD and �lim have to be defined.

The definition of the parameter FD depends on the memory surface F . For astress path going through the memory surface, this parameter converges to its maximalvalue. For a stress point on the memory surface, the parameter is reduced to zero dur-ing cyclic loading:

FD ��

PDd�vp � �44�


244

• stress point within the memory surface and �v � k0:

PD � P�FD�max � FD�

FD�maxFL �45�

with FD�max fitparameter,• stress point on the memory surface and �v � k0:

PD � �0�01PFD

FD�max� �46�

• stress point on the memory surface and �v k0:

PD � 0�0 � �47�

The parameter �lim is defined as a limitation of the distance between the loading andthe bounding surface. For values � � �lim, an additional isotropic hardening of thebounding surface occurs. For � � �lim, this additional hardening of the bounding sur-face decreases. The plastic modulus of the additional isotropic deformation of thebounding surface is given by:

�PZ � P FS�FB�max � FB�

FB�max

� �0�2

for � � �lim � �48�

�PZ � �PFB

FB�max�0�001 � FS� for � � �lim �49�

with

FB�max � ��k1FS � FD 2�0�FL � �50�


245

Figure 10.27: Multiple-step test.


246

Figure 10.28: Path 03, �B �7.1‰.

Figure 10.29: Path 07, �B �2.8‰.

Figure 10.30: Path 07, �B �4.9‰.


247

Figure 10.31: Path 08, �B �4.9‰.

Figure 10.32: Path 09, �B �2.8‰.

Figure 10.33: Path 09, �B �4.9‰.

FB ��

�PZd�vp � �51�

�lim � 1�0 ��k1F0�2S � FD 2�0� � �52�

�PZ plastic modulus for the additional isotropic hardening of the bounding surface dueto non-proportional loading,

��k material parameter,FB 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 theamount 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 resultsof 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 construc-tions like necked girders, girders with holes, or plates with holes. Figures 10.34 to10.36 show specimens used in these investigations.

All experiments were performed force-controlled. The longitudinal strains in theinteresting sections were measured with strain gauges.

For the description of the structural behaviour, the Finite-Element method wasused. Precise informations concerning details of the computational methods can befound 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 calcula-tions are presented. The cyclically loaded girder (Figure 10.35) shows repeating plasticdeformations in the area of the neck.


248

10.4 Experiments on Structural Components

249

Figure 10.34: Plate with a hole.

Figure 10.35: Necked girder.

In the right diagram of Figure 10.37, the force-strain relation measured directly atthe edge of the neck is presented. The force-strain relation predicted by the ETS-modelis plotted in the left diagram of this figure.

The calculated force-strain relation shows a good correspondence with the resultsof the experiment. The amount of the increase of the plastic strains per cycle is wellpredicted 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 andcalculated strains in the area of the lower turnover points is caused by the more inten-sive bulge of the calculated hysteresis-loops.


250

Figure 10.36: Girder with holes.

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

As a second example, Figure 10.38 shows a comparison between measured and cal-culated strains �xx of a plate with a hole (X � 0�0 in Figure 10.34) for the first 10 loadsteps. It is seen that the differences between the results of the calculations with the modelsof Reininghaus (biaxial) [2] and Scheibe (uniaxial, Z � 0) [1] are small. As an additionalresult, Figure 10.38 shows that the “biaxial” extensions of the ETS-model to include non-proportional effects do not influence the results of the calculations of the uniaxial loadedplate with a hole. If a non-proportional load occurs, distinct differences are obtainedmerely 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 be-haviour are necessary for a precise calculation of the response of structures under plas-tic cyclic loads.

Extensive investigations were carried out into the material behaviour of structuralmild steel Fe 510 under uniaxial, load- and strain-controlled loads as well as under dif-ferent biaxial proportional and non-proportional loads combined with yield-surface in-vestigations.

10.5 Summary

251

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

��

The two-surface model of Dafalias-Popov was extended to fit the individual char-acteristics of mild steel behaviour under cyclic loads. This extended-two-surface modelis able to simulate precisely the behaviour of Fe 510 under various proportional andnon-proportional multiaxial cyclic loads.

Linking up with a Finite-Element program, structures and structural elements un-dergoing cyclic or random plastic deformations are calculated.

References

[1] H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Berucksichtigungin Konstruktionsberechnungen. Tech. Univ. Braunschweig, Institut fur Stahlbau, BerichtNr. 6314, 1990.

[2] M. Reininghaus: Baustahl Fe 510 unter zweiachsiger Wechselbeanspruchung. Tech. Univ.Braunschweig, Institut fur 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 Veranderung der Fließflache von Baustahl bei mehrachsiger plas-tischer Wechselbeanspruchung. Dissertation TU Braunschweig, 1999.

[5] M. J. Michno, W. N. Findley: A Historical Perspective of Yield Surface Investigations forMetals. Int. J. Non-Linear Mechanics 11 (1976) 59–82.

[6] D. Besdo, N. Wellerdick-Wojtasik: The Influence of Large Torsional Prestrain on the Tex-ture Development and Yield Surfaces of Polycrystals. This book (Chapter 7).

[7] H. Heuer: Untersuchung zur Anwendbarkeit des Einfließflachen-Modells auf das zyklischeMaterialverhalten von Baustahl. Diplomarbeit, Tech. Univ. Braunschweig, 1988.

[8] Y. F. Dafalias, E.P. Popov: A Model of Nonlinearly Hardening Materials for Complex Load-ings. Acta Mechanica 21 (1975) 173–192.

[9] J. Scheer, H.-J. Scheibe, D. Kuck, M. Reininghaus: Stahlkonstruktionen unter zyklischer Be-lastung. Arbeits- und Ergebnisbericht 1987–1990. Subproject B5, Collaborative ResearchCentre (SFB 319): Stoffgesetze fur das inelastische Verhalten metallischer Werkstoffe – Ent-wicklung und Technische Anwendung, Tech. Univ. Braunschweig, Institut fur Stahlbau,1990.

[10] U. Peil, D. Kuck: Stahlkonstruktionen unter zyklischer Belastung. Arbeits- und Ergebnisbe-richt 1991–1993. Subproject B5, Collaborative Research Centre (SFB 319): Stoffgesetze furdas inelastische Verhalten metallischer Werkstoffe – Entwicklung und Technische Anwen-dung, Tech. Univ. Braunschweig, Institut fur Stahlbau, 1993.

[11] U. Peil, M. Reininghaus: Baustahl unter mehrachsiger zyklischer Belastung. Arbeits- undErgebnisbericht 1991–1993. Subproject B5, Collaborative Research Centre (SFB 319):Stoffgesetze fur das inelastische Verhalten metallischer Werkstoffe – Entwicklung und Tech-nische Anwendung, Tech. Univ. Braunschweig, Institut fur Stahlbau, 1993.

[12] M. Reininghaus: Baustahl ST52 unter plastischer Wechselbeanspruchung. Dissertation TUBraunschweig, 1994.

[13] Z. Mroz: An Attempt to Describe the Behavior of Metals under Cyclic Loads Using a MoreGeneral Workhardening Model. Acta Mechanica 7 (1968) 199–212.

[14] J. Scheer, H.-J. Scheibe, D. Kuck: Zum Verhalten ausgeklinkter Trager unter zyklischerBeanspruchung. Bauingenieur 65 (1990) 463–468.


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11 Theoretical and Computational Shakedown Analysisof Non-Linear Kinematic Hardening Materialand Transition to Ductile Fracture

Erwin Stein, Genbao Zhang, Yuejun Huang, Rolf Mahnkenand Karin Wiechmann*

Abstract

The research work of this project is based on Melan’s static shakedown theorems forperfectly plastic and linear kinematic hardening materials. Using a 3-D generalizationof Neal’s 1-D model for limited hardening, a so-called overlay model, a new theoremand a corollary are derived for general non-linear kinematic hardening materials. Forthe numerical treatment of the structural analysis of 2-D problems, the Finite-Elementmethod (FEM) is used, while enhanced optimization algorithms are used to perform theshakedown analysis effectively. This will be demonstrated with some numerical exam-ples. Treating a crack as a sharp notch, the shakedown behaviour of a cracked ductilebody 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 verycomplicated. If the applied loads are small enough, the system will remain elastic forall possible loads. Whereas if the ultimate load of the system is attained, a collapsemechanism will develop and the system will fail due to infinitely growing displace-ments. Besides this, there are three different steady states that can be reached, while theloading proceeds: 1. Incremental failure occurs if at some points or parts of the system,

253

* Universitat Hannover, Institut fur Baumechanik und Numerische Mechanik, Appelstraße 9 a,D-30167 Hannover, Germany



the remaining displacements and strains accumulate during a change of loading. Thesystem will fail due to the fact that the initial geometry is lost. 2. Alternating plasticityoccurs, this means that the sign of the increment of the plastic deformation during oneload cycle is changing alternately. Though the remaining displacements are bounded,plastification will not cease and the system fails locally. 3. Elastic shakedown occurs ifafter initial yielding plastification subsides and the system behaves elastically due tothe fact that a stationary residual stress field is formed and the total dissipated energybecomes stationary. Elastic shakedown (or simply shakedown) of a system is regardedas a safe state. It is important to know if a system under given variable loadings shakesdown 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 hyper-static systems consisting of elastic, perfectly plastic materials. This theorem was thengeneralized by Melan [2, 3] in 1938 to continua with elastic, perfectly plastic and lin-ear unlimited kinematic hardening behaviour. Koiter [4] introduced a kinematic shake-down theorem for an elastic, perfectly plastic material in 1956, that was dual to Me-lan’s static shakedown theorem. Since then, extensions of these theorems for applica-tions of thermoloadings, dynamic loadings, geometrically non-linear effects and internalvariables have been carried out by different authors (Corradi and Maier [5], Konig [6],Maier [7], Prager [8], Weichert [9], Polizzotto et al. [10]). However, little progress hasbeen made in the formulation of a corresponding shakedown theorem for materialswith non-linear kinematic hardening. The only attempt was made by Neal [11], whoformulated a static shakedown theorem for materials with non-linear kinematic harden-ing in a 1-D stress state by using the Masing overlay model [12]. Several papers werepublished concerning especially 2-D and 3-D problems (Gokhfeld and Cherniavsky[13], Konig [14], Sawczuk [15, 16], Leckie [17]). The shakedown investigation ofthose problems leads to grave mathematical problems. Thus, in most of these papers,approximate solutions based on the kinematic shakedown theorem of Koiter [4] or onthe assumption of a special failure form were derived. But these solutions often losttheir bounding character due to the fact that simplifying flow rules or wrong failureforms were estimated. Until the beginning of project B6, only a few papers were pub-lished, in which the Finite-Element method was used for the numerical treatment ofshakedown 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 oflinear elastic, perfectly plastic, of linear elastic, linear unlimited kinematic hardeningand of linear elastic, non-linear limited kinematic hardening materials were taken under

11 Theoretical and Computational Shakedown Analysis

254

consideration. The theoretical and numerical treatment of the shakedown behaviour ofthese material models was one major scope of project B6. Based on static shakedowntheorems, the numerical treatment of 2-D and 3-D field problems for arbitrary non-lin-ear kinematic hardening materials by Finite-Element method should be realized. Onespecial task was the formulation and the proof of a static shakedown theorem for lim-ited non-linear kinematic hardening materials. In Section 11.2, a 3-D overlay model ispresented, that was developed to describe non-linear, limited kinematic hardeningmaterial 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 theproposed material model, are extensions of Melan’s static shakedown theorems for per-fectly plastic and linear kinematic hardening materials [2].

While analytical solutions of shakedown problems can only be derived for verysimple systems, Finite-Element methods based on displacement methods should beused 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 anon-linear mathematical optimization problem is derived, that is very large scaled.Solving optimization problems like these is normally very difficult. Thus, effective opti-mization algorithms should be formulated and implemented, that were designed especi-ally to take account of the special structure of the problems. Section 11.3 is concernedwith the numerical approach based on static shakedown theorems. The discretized opti-mization problems for the proposed material models are discussed briefly.

In Section 11.3.5, numerical examples show the effectivity of the proposed formu-lation. Solutions for perfectly plastic and kinematic hardening materials are compared.One important scope of project B6 was the examination of hardening and softeningmaterials. While classic shakedown theorems imply implicitly that a material under cyc-lic loading behaves stable after only one or two loading cycles, experimental investiga-tions show that stable cycles can be reached only after several loading cycles andsometimes only asymptotically. Thus, the influence of cyclic hardening and softeningon the shakedown behaviour of materials had to be taken under consideration. The ex-amination 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 materialis carried out, and the results are compared with those of the 3-D overlay model de-scribed in Section 11.2.

Stress singularities occur if macroscopic cracks develop in a solid material. Underthese circumstances, classical shakedown theorems cannot be used. Thus, one majoraim of project B6 was to apply shakedown theory directly to fatigue fracture to includestress singularities into shakedown investigations. In Section 11.4, we will apply shake-down theory to fatigue fracture to derive thresholds for no crack propagation. Classicshakedown theorems predict a zero shakedown limit load for a cracked body becauseof singular stresses at the crack tip. But experiments for ductile materials show thatlimits exist, for which no crack propagation occurs. We will consider the crack as asharp 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 plasticenergy dissipated in the cracked body.

1.1 Introduction

255

11.2 Review of the 3-D Overlay Model

There exist many mathematical models to describe the kinematic hardening behaviourof materials, for example the Prager linear kinematic hardening model [25], its modifi-cation by Ziegler [26], Mroz multisurface model [27], Dafalias and Popov’s two-sur-face model [28] and so on. In Stein et al. [29], a so-called 3-D overlay model was de-veloped to describe the non-linear kinematic hardening material behaviour. We willgive here a brief review of the proposed model.

A macroscopic material point x � � � IR3 is assumed to be composed of a spec-trum of microscopic elements (or microelements). Each microelement is numbered witha scalar variable � � �0� 1�. Stresses and strains are separately defined for the macro-scopic material point (macrostress and macrostrain) and the microelements (micro-stresses and microstrains). They are denoted by ��x�� x� and ��x� �� x� ��, respec-tively.

The macrostress � has to fulfil the equilibrium condition:

div��x� � b�x�� x � � � �1�

In the framework of geometrically linear continuum mechanics, the kinematic relation

��x� � 12�u�x� �u�x��T�� x � � � �2�

holds between the displacement u and the strain �.By assuming that the stress � of a macroscopic material point (macrostress) is the

weighted sum of the stresses � of all microelements (microstresses), the macroscopicmaterial point and the corresponding microelements deform in the same way, and weget the following static and kinematic relations:

��x� ��1

0

��x� ��d� � �3�

��x� �� x�� 0� 1� � �4�

Furthermore, we suppose that all microelements are linear elastic, perfectly plastic andhave the same temperature, the same elastic moduli, but different yield stresses k��.For convenience, k�� can be considered as a monotone growing function of �. Addi-tionally, the validity of the additive decomposition of the microstrain � in an elasticand a plastic part is supposed. Thus, the following relations can be derived for the mi-croelements:

�� E�� P�� 5�


256

�E�� E�1�� 6�

�� k2�� 7�

��P��

� �� 0 � �8�

�� k2�� 0 � �9�

where E stands for the symmetrical elasticity tensor and �� is the yield function. Forsimplicity, the argument x of the fields will be omitted partly.

Assume that at a certain macrostress, the microelement number � begins to yield,the function k�x� �� is then uniquely determined by the macroscopic �� -function in the1-D case:

��

0

k��d�� 1 � ��k�� 10�

It is easy to show that, similar to k�� is also a monotone growing function of thevariable � (see Figure 11.1). For � � 0, we have � � �� 0� � k�� 0� � k0. Themaximum value of � (denoted by �Y or K) is derived by setting � � 1 in Equation(10):

�Y � �� 1� � K ��1

0

k��d� � �11�

Thus, k0 and K can be identified with the initial yield stress �o and the ultimate stressof the macroelement �Y, respectively. For the 3-D case, there is an analogous relationbetween � and k�� as Equation (10), namely:

11.2 Review of the 3-D Overlay Model

257

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

��

��0

k��d�� 1 � ��k�� 12�

Now, we define the difference between the microstress �� and the macrostress � asthe residual microstress and denote it by ��:

�� 13�

For the residual microstress defined in this way, we have:

�� 0� � � �0� 1�� for �� o � �14�

and

�� 0� � � �0� 1�� for �� o � �15�

It is necessary to notice that Equation (15) holds for all � � �0� 1� even if �� is smal-ler than k��.

Integration of Equation (13) yields:

�1

0

��x� ��d� � 0� � x � � � �16�

where Equation (3) has been used. From Equation (13) can be concluded that the resul-tant of �� does not contribute to the macrostress �. All microstress fields, which ful-fil Equation (16), can be regarded as residual microstress fields.

For a kinematic hardening material, it is possible to represent the residual micro-stress ��x� �� by a backstress��x�, which describes the translation of the initial yieldsurface:

��x� �� k�� k�0��x� � �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 theo-

rem has been formulated and proved (e.g. Stein et al. [30], Zhang [31]):

Theorem 1: If there exist a time-independent residual macrostress field ��x� and atime-independent residual microstress field ��x� ��, satisfying

��m��x� 0�� K�x� � k0�x��2 � � x � � � �18�

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


258

��m��E�x� t� ��x� ��x� 0�� k20�x� �19�

is fulfilled � x � � and � t � 0, where m � 1 is a safety factor against inadaptation, thenthe total plastic energy dissipated within an arbitrary load path contained within the loaddomain 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 �. Con-sidering the relation between the backstress �� and the residual microstress � in Equation(17), it is also possible to formulate the following corollary directly in terms of the back-stress ��, i.e.:

Corollary 1: If there exist a time-independent residual macrostress field ��x� and atime-independent field �� x�, satisfying

��m �� x�� K�x� � k0�x��2 � � x � � � �20�

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

��m��E�x� t� ��x� � �� x�� k20�x� �21�

is fulfilled � x � � and � t � 0, then the system will shake down.

For the formulation of the static shakedown theorem 1 and the corresponding corollary1, only the values of k0 and K have been used. That means that the shakedown limitsfor systems of the considered material do not depend on the particular shape of thefunction k��, and therefore do not depend on the particular �� -curve but, solely, onthe magnitudes of k0 and K.

For K�x� � k0�x� (an elastic, perfectly plastic material), we have �� 0 due toEquation (20), and theorem 1 reduces to Melan’s theorem [2, 3] for an elastic, perfectlyplastic material. For K�x� � � (materials with unlimited kinematic hardening), theconstraint (Equation (20)) imposed on the backstresses �� x�, can never become activeand therefore may be dropped. In this case, we get the static shakedown theorem ofMelan 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 kinematictheorems can be employed. However, the use of the static theorems has the advantagethat the discretized optimization problem is regular. Therefore, only static theoremswere used for the following considerations. Furthermore, for the Finite-Element discre-


259

tization, only elements based upon the displacement method were used. In the follow-ing sections, we will limit our attention to plane-stress problems (for more details andother 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 al-lowing still for shakedown. Thus, for systems consisting of linear elastic, perfectly plas-tic material, we get the following optimization problem in matrix notation:

� � max � �22�

�NGi�1

Ci�i � C� � 0 � �23�

��Ei �j� �i� � �2

o � ��i� j� � � � � � �24�

for load domains of the form of a convex polyhedron with M load vertices. Here, � isthe maximal possible enlargement of the load domain, NG is the number of Gaussianpoints. � � �1� � � � �NG� and � � �1� � � � �M� are sets containing all Gaussian points andload vertices, respectively. C is a system-dependent matrix and �E

i �j� is the elasticstress 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 equilib-rium conditions for the residual stresses and the shakedown conditions controlled at theGaussian points, respectively. �i is the residual stress vector in the i-th Gaussian point.

In general, the discretized shakedown problem (Equations (22) to (24)) is a largescaled optimization problem. Direct use of standard optimization algorithms such as thesequential quadratic programming method (SQP-method) is not effective. Thus, two op-timization algorithms were formulated and implemented to take account of the specialstructure 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 attackan alternative problem, the dual problem, whose unknowns are the Lagrange multi-pliers of the primal problem.

In order to solve the dual problem, a projection method is used. Each subproblemis then solved iteratively. The iteration matrix needed to do so was implemented due toBertsekas’s algorithm. To make the algorithm even more effective for large sized prob-lems, 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

11.3.2.2 A reduced basis technique

The main idea is to solve Equations (22) to (24) in a sequence of reduced residualstress spaces of very low dimensions. Starting from the known state ��k�1� and ��k�1�,a few �r� basis vectors bp�k� p � 1� � � � r, are selected from the residual stress space �d.They form a subspace �r�k (or reduced residual stress space) of �d. The improved state�k� is determined by solving the reduced optimization problem. Due to its low dimen-sion, the reduced problem can be solved very efficiently by using a SQP- or a penaltymethod. The k-th state is obtained by an update for �k and ��k. Selecting new reducedbasis vectors, the process is repeated until a convergence criterion for ��k is fulfilled.One way for generating the basis vectors is to carry out an equilibrium iteration. Theintermediate 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 ki-nematic hardening material, we restrict the load domain in the same way as in Section11.3.1, and the following optimization problem can be formulated [31]:

� � max � �25�

�NGi�1

Ci�i � 0 � �26�

��Ei �j� �i ��i� � �2

o � ��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 � �i ��i � �28�

where the yi are not restricted. Thus, the equality constraints (Equation (26)) of Equa-tions (25) to (27) can be dropped:

� � max � �29�

��Ei �j� yi� � �2

o � ��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

Lemma 1: The maximal shakedown load factor �s of Equations (29) and (30) can bederived through

�s � �� with �� mini��

�i � �31�

where �i, �i � ��, is the solution of the subproblem

�i � max � �32�

��i�Ei �j� yi� � �2

o � �33�

The dimension of Equations (32) and (33) is very low. Thus, it can be solved effec-tively with a usual SQP-method.

Relationship between perfectly plastic and kinematic hardening materialThe shakedown load of a system consisting of unlimited kinematic hardening materialis determined by that point xip , where the maximum possible enlargement of the elasticstress domain �E

ip is the least in comparison to other points. Thus, the failure is of localcharacter. This reflects the fact that a system, that consists of unlimited kinematic hard-ening material and is subjected to cyclic loading, can fail only locally in form of alter-nating plasticity [34]. Incremental collapse cannot occur since it is connected with anon-trivial, kinematic compatible plastic strain field, which has global character.

The shakedown load of a system consisting of perfectly plastic material cannot belarger than the shakedown load of the same system consisting of unlimited kinematichardening material with the same initial stress �o. However, it is possible that theshakedown 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 toshow that

�ip � yip �34�

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

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

2. The kinematic hardening does not influence the shakedown load if the same sys-tem with perfectly plastic material, subjected to the same loading, fails in form ofalternating plasticity in such a manner that there exists at least one point xip , forwhich the enlarged elastic stress domain �s�E

ipis just contained in the yield surface

shifted to ��ip . A further shift of the yield surface at this point would cause that aportion of the enlarged elastic domain � � s�E

ip leaves the yield surface. In the se-quel, the alternating plasticity with the special character mentioned before will bedenoted by APSC. In all other cases, an increase of the shakedown load due to ki-nematic hardening is expected.


262

3. If the shakedown loads for perfectly plastic and unlimited hardening material areidentical, 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 identi-cal with that determined for unlimited kinematic hardening material, provided thelatter 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 kine-matic hardening material depends only on the values k0 and K. For this reason, the givenfunction k�� may be replaced by a step function of � � �0� 1�. The step function has to bechosen such that its minimum is equal to k0 and its area is equal to K. That is,

K ��ml�1

��l �kl � m 2 � �35�

where m is the number of the intervals of the step function and �kl is the value of thestep function of the l-th interval with the length ��l.

For plane stress problems, the microelements may be incorporated in a naturalway. The thickness t of the structure is divided into several (m 2) layers with thethicknesses tl, l � 1� 2� � � � �m. Each layer behaves elastic, perfectly plastic and has acorresponding yield stress kl (one value of the step function). The thicknesses of thelayers have to be chosen such that

tlt� ��l � � l � �1� � � � �m� � �36�

By doing so, a unique relation between the layers of the structure and the intervals ofthe step function is established. The microelements of the l-th interval are replaced bythe l-th layer of the structure. The parallel connection of the microelements is realizedby discretizing all layers in the same way. That is, elements that lie on top of eachother have the same nodes. Thus, it is guaranteed that elements of different layers havethe 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]):

� � max � �37�

�NGi�1

Ci

�ms�1

tst�i�s �

�NGi�1

Ci�i � C� � 0 � �38�

��Ei �j� �i�1� � �k2

1 � k20 � � �i� j� � � � � � �39�


263

��i�1� � �K � �k1�2 � �K � k0�2 � � i � � � �40�

The Equations (38), (39) and (40) represent the discretized static equilibrium conditionsfor the residual macrostresses and the shakedown conditions controlled at the Gaussianpoints, respectively. �i�s and �i�s are the residual stress vector and the residual micro-stress vector in the s-th layer of the i-th Gaussian point. Between �i�s� �i�s and the resi-dual macrostress vector �i, the following relation holds:

�i�s � �i �i�s � �41�

For m � 1 and K � �k1, Equations (37) to (40) reduce to the discretized shakedownproblem for a perfectly plastic material. In this case, we have �i�1 � 0� �i � �i�1 and theconstraint (Equation (40)) can be dropped. We come to the other extreme case by as-suming K � �, which corresponds to an unlimited kinematic hardening material. Dueto K � �, the constraint (Equation (38)) can never become active, and thus �i�1 is notconstrained. Correspondingly, the constraint (Equation (40)) may be dropped as well.

To solve the optimization problem (Equations (37) to (40)) effectively, the re-duced 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 influenceof 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 pres-sure p and an internal temperature Ti (Figure 11.2). The external temperature Te isequal to zero for all times t. For Ti � Te and p � 0, the system is assumed to be stress-free. The pressure and the temperature can vary between zero and their maximum val-ues �p and �Ti. The corresponding load domain is defined by:

0 � p � ��1po � �p � 0 � �1 � 1 � 0 � Ti � ��2Toi � �Ti � 0 � �2 � 1 � �42�

To visualize the influence of kinematic hardening on the shakedown limits, the follow-ing three constitutive laws are considered:

1. Elastic, perfectly plastic material with initial yield stress ko (curve 1 in Figure11.2 b).

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

3. Unlimited kinematic hardening with K � � (curve 3 in Figure 11.2 b).


264

Curves 1, 2 and 3 in Figure 11.2b show that the kinematic hardening does not alwaysincrease 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 al-ternating 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 pro-file with a cope on either side. The girder is simply supported and, in the middle, it issubjected to a concentrated single load P. Both corners of the copes are provided witha round drill hole of radius r � 8 mm (see Figure 11.3a). The material is St 52-3 withan initial yield stress �o � 37�5 kN/cm2 and a maximal hardening �Y � 52�0 kN/cm2.The hardening can be regarded as kinematic.

Experimental investigationsAt the Institute for Steel Construction of the University of Braunschweig, the girderwas investigated experimentally [35]. First of all, the behaviour of the system subjectedto cyclic loading was of interest. Additionally, for comparison, the ultimate load wasdetermined experimentally as well.

Firstly, the girder was subjected to different cyclic load programs. The load pro-gram of the first 15 cycles is shown in Table 11.1. Then, the girder was subjected to aload 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 drillhole, 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 by50 cm, and it was recoped as before. Then, for this system, the ultimate load was deter-mined as 887 kN. Note that this value can also be regarded as the ultimate load of theinitial system since only those cross-sections near the cope are responsible for failureof the system.


265

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

Numerical investigationsDue to the symmetry of the system and the loading, only one quarter of the system wasconsidered for numerical investigation. For the Finite-Element discretization, two differenttypes of elements were employed. The web of the girder was discretized by using 8128isoparametric elements each with 4 nodes (see Figure 11.3 a). The upper and the lowerflanges were discretized with 768 and 640 DKT (discrete Kirchhoff triangle)-elements,respectively. Apart from bending forces, the DKT-elements can also be stressed in planein 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 loadwere calculated. The solutions were obtained by the reduced basis technique. In orderto demonstrate the influence of kinematic hardening on the ultimate load and on theshakedown load, respectively, the calculations were performed both for a perfectly plas-tic 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) was1% 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�Y��o due to kinematic hardening, the shakedown load remains unaltered. In this case, thegirder fails due to alternating plasticity near the drill holes (see Section 11.3.2.1).


266

Figure 11.3: a) Discretization of the steel girder; b) load-strain diagram at one of both drill holesof 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

It should be pointed out that, originally, the experiment was not intended to ana-lyse shakedown behaviour. During the experiment, the amplitudes of the cyclic loadswere higher than the theoretical shakedown load. Thus, no shakedown behaviour couldbe observed. However, some valuable information can be drawn from the load-straindiagram for the first 5 load cycles shown in Figure 11.3 b. The strains were measureddirectly at one of the drill holes.

In Figure 11.3 b, a region can be observed, where the load P is linearly propor-tional to the strains, i.e., where the system behaves purely elastic. The amplitude of theregion is between 155.2 kN and 179.5 kN. A comparison shows that the numerical re-sults are in good agreement with the experiment.

11.3.5.3 Incremental computations of shakedown limits of cyclic kinematic hardeningmaterial

To describe the cyclic kinematic hardening behaviour of materials, many models havebeen developed. Mroz’s multisurface model [27], the two-surface model of Dafaliasand 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 behav-iour. Due to the fact that no shakedown theorems for cyclic hardening materials havebeen formulated yet, an incremental method will be used to calculate the shakedown limit.

Examples and comparisonAs our first example, we consider the square plate with circular hole illustrated in Fig-ure 11.4. The length of the plate is L and the ratio between the diameter D of the holeand the length of the plate is 0.01. The thickness of the plate is t � 1 cm. The systemis subjected to the biaxial loading p1 and p2. Both can vary independently betweenzero and their maximum values �p1 and �p2. The corresponding load domain is definedby:

0 � p1 � ��1�o � �p1 � 0 � �1 � 1 � �43�0 � p2 � ��2�o � �p2 � 0 � �2 � 1 � �44�

The results are shown in Table 11.3, where �pih denotes the shakedown limit for thepath-independent hardening material and �cyh the shakedown limit for the cyclic kine-matic hardening material.


267

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 633.2 164.22. Kinematic hardening 877.7 164.2

It is shown that the results from optimization methods with cyclic independent hard-ening properties are upper bounds for those from incremental computation with cyclickinematic hardening materials. It turned out that for about 20 cycles, the values �cyhapproach �pih with an error less than 1%. It should be pointed out that the computationefforts 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 inFigure 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 ofshakedown limits. Table 11.4 shows the results of the incremental computations.

For this example too, the shakedown limits are the same as those of numericaloptimization apart from the fact that the number of load paths, which we have takenfor the incremental hardening, has no influence on the shakedown limit of the system.

Remark 1: Elastic shakedown does not implement damage and creep phenomena. En-gineers are interested in the admissible number of cycles for low-cycle fatigue,which cannot be derived from classical shakedown theorems. An approximatedreduced load factor �� for scalar-valued damage can be achieved in a post-process assuming conservatively that the loads always alternate between theirmaximum and minimum values in all load cycles.


268

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.

��

0.0/1.0 0.69633 0.69629 3000.2/1.0 0.65458 0.65417 200.4/1.0 0.61758 0.61750 200.6/1.0 0.58433 0.58417 200.8/1.0 0.55471 0.55417 201.0/1.0 0.52775 0.52707 20

Remark 2: There is an important connection between shakedown theory and structuraloptimization admitting inelastic deformations, which is a demanding task in theframe of the design under modern safety considerations related to the failure of astructure 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 complicatedgeometry and load domain, the numerical methods like Finite-Element method should beusually used (see [32, 36]). For a problem with no more than two loading parameters (loaddomain with four vertices), Stein and Huang [37] developed an analytical method for de-termination of the shakedown-load factor �. The advantage of this method is that only themaximum effective stress in the system must be calculated. The shakedown-load factor �follows directly from a closed form. For a load domain with only one parameter (loaddomain with two vertices), the result for � is especially simple, it reads:


269

Figure 11.5: Geometry and loading conditions of a compact tension specimen.

Table 11.4: Shakedown limits for a CT-specimen.

��

� � �� 0.069325 0.069292 20� � �� 0.085133 0.085083 20� � �� 0.092504 0.092083 20� � �� 0.101117 0.101042 20� � �� 0.107392 0.107292 20

� � 2�o

�eff� �45�

where �eff is von Mises effective stress and �o is the elastic limit. From Equation (45)can be concluded that the shakedown limit load of the system is as large as twice as itselastic limit.

Making use of this result, Huang and Stein [38] calculated the shakedown limitload of a notched body under variable tension � as illustrated in Figure 11.6 a. It reads:

� � 2�o

�eff� �o

��r

�

K��1 � 2

� � �46�

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

Ksh � �K � �o��r

��1 � 2

� � �47�

If the applied SIF K does not exceed the shakedown limit SIF Ksh, the notched bodyshakes down. Otherwise, alternating plasticity occurs at the notch root.

Shakedown limit SIF for a cracked bodyIn [38], Huang and Stein applied Neuber’s material block concept [24] to the shake-down investigation of a cracked body. Accordingly, the continuum ahead of a sharpnotch is considered as a material block with finite linear dimension � (Figure 11.6 b).Across this block, no stress gradient can develop. The original notch should be re-placed by an effective notch with radius r� �� r�. The stress-concentration factor is re-


270

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

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 theblock � of the original notch is equal to the maximum stress of the modified notch, i.e.:

�ymax �r� �1�

��

0

�y�rdr1 � �48�

Using Craeger’s relation for the stress distribution [39] at the notch root, the followingrelation can be derived:

2K��r�� 1

�

��

0

K��2�r�2 r1�

� 1 r2�r�2 r1�

� �dr1 � 2K��

�r 2�� 49�

For r�, one gets:

r� � r 2� � �50�

Thus, the effective notch radius is equal to the original one plus two times Neuber’smaterial block size �.

In the case of a crack (Figure 11.6 c), the effective crack-tip radius, denoted by �, canbe obtained immediately by setting r� � r 2� and r � 0 in Equation (47), yielding:

� � 2� � �51�

Equation (51) indicates implicitly that a crack can be treated as a notch with radius �.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 anidentical manner as a notched body does if the latter has the same geometry as thecracked body and the notch root radius r is small enough (see also [43, 44]). Theshakedown limit stress-intensity factor reads:

Ksh � �o��

��1 � 2

� � �52�

Thus, the shakedown limit stress-intensity factor of a cracked body is proportional tothe initial yield stress �o times the square root of the effective crack-tip radius �. For amaterial, �o is usually given. The problem now is to establish the effective crack-tip ra-dius �. A direct measurement of this parameter is difficult. Using an indirect method,Kuhn and Hardrath [45] calculated the effective crack-tip radii for metallic materialsand proposed a relationship between � and the ultimate strength �Y of a material de-picted in a diagram.

For a given material, the initial yield stress �o and ultimate stress �Y can be mea-sured by a simple tension experiment. The effective crack-tip radius � is taken directlyfrom the diagram given in [45]. Knowing these quantities, the shakedown limit SIF Ksh


271

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 samematerials are listed there. They were obtained by other authors with experimental meth-ods of other theoretical approaches [46, 47].

It can be seen that the shakedown limit SIFs Ksh of cracked bodies agree quitewell with their fatigue thresholds Kth. This agreement indicates that the reason forcrack 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-lin-ear hardening materials. For this class of materials, a static shakedown theorem and acorresponding corollary were formulated and proved, which are generalizations of Me-lan’s static shakedown theorems for perfectly plastic and linear kinematic hardeningmaterials. A systematic investigation of the numerical treatment of shakedown prob-lems using Finite-Element method was carried out. The findings were used to employefficient optimization strategies and algorithms developed to take advantage of the spe-cial structure of the arising optimization problems. As an important result of these in-vestigations, explicit conclusions about the failure forms of cyclically loaded systemscould be drawn, i.e. incremental collapse or alternating plasticity. The influence of cyc-lic hardening and softening on the shakedown behaviour of structures was studied in-crementally. The results were compared with those derived for the 3-D overlay model.

A new methodology was proposed to include stress singularities into shakedowninvestigations allowing for the prediction of fatigue thresholds of ductile crackedbodies. Thus, a transition from shakedown theory to cyclic fracture mechanics wasachieved.


272

Table 11.5: Shakedown limit SIFs �� and fatigue thresholds � � for various materials.

Material �� [MPa] �� [MPa]��

�m1/2 �� [Mnm–3/2] � � [Mnm–3/2] Ref.

2 1/4 Cr-1Mo 345 528 13.549 · 10–3 8.25 8.3 [45]SA 387-2.22 390 550 12.433 · 10–3 8.6 8.3 [45]SA 387-2.22 340 520 13.708 · 10–3 8.14 8.6 [45]SA 387-2.22 290 500 14.346 · 10–3 7.3 9.6 [45]Docol 350 260 360 19.128 · 10–3 8.9 5.4 [46]SS 141147 185 322 22.316 · 10–3 7.6 6.0 [46]HP Steel 210 304 25.504 · 10–3 9.5 6.2 [46]HP Steel 160 279 28.692 · 10–3 8.1 6.7 [46]HP Steel 120 242 36.662 · 10–3 7.8 8.2 [46]

References

[1] H. Bleich: Über die Bemessung statisch unbestimmter Stabwerke unter der Berucksichti-gung des elastisch-plastischen Verhaltens der Baustoffe. Bauingenieur 13 (1932) 261–267.

[2] E. Melan: Der Spannungszustand eines Mises-Henckyschen Kontinuums bei veranderlicherBelastung. Sitzber. Akad. Wiss. Wien IIa 147 (1938) 73–78.

[3] E. Melan: Zur Plastizitat des raumlichen 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. Konig: 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 Tempera-

ture. In: Proc. Symp. Plasticita nella Scienza delle Costruzioni, Bologna, 1956.[9] D. Weichert: On the Influence of Geometrical Nonlinearities on the Shakedown of Elastic-

Plastic 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 elasti-

sche Spannungen. Technical Report 3, Wissenschaftliche Veroffentlichungen aus dem Sie-mens Konzern, 1924.

[13] D.A. Gokhfeld, O.F. Cherniavsky: Limit Analysis of Structures at Thermal Cycling. Sijt-hoff & Noordhoff, 1980.

[14] J.A. Konig: 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 IUTAMSymp. 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 Struc-tures. Comp. Math. Appl. Mech. Eng. 3 (1974) 37–53.

[20] J. Gross-Weege: Zum Einspielverhalten von Flachentragwerken. PhD Thesis, Inst. furMech., Ruhr-Universitat Bochum, 1988.

[21] H. Nguyen Dang, P. Morelle: Numerical Shakedown Analysis of Plates and Shells of Revo-lution. 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 plasti-schem Material. PhD thesis, Inst. fur Computeranwendungen, Universitat 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.

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[25] W. Prager: A New Method of Analyzing Stresses and Strains in Workhardening Plastic Sol-ids. 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. Mroz: 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 Load-ing. Acta Mechanica 43 (1975) 173–192.

[29] E. Stein, G. Zhang, R. Mahnken, J.A. Konig: Micromechanical Modeling and Computationof 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. Konig: Shakedown with Nonlinear Hardening Including StructuralComputation Using Finite Element Method. Int. J. Plasticity 8 (1992) 1–31.

[31] G. Zhang: Einspielen und dessen numerische Anwendung von Flachentragwerken aus idealplastischem bzw. kinematisch verfestigendem Material. PhD thesis, Institut fur Baumechanikund Numerische Mechanik, Universitat Hannover, 1992.

[32] E. Stein, G. Zhang, R. Mahnken: Shakedown Analysis for Perfectly Plastic and KinematicHardening Materials. In: Progress in Computational Analysis of Inelastic Structures,Springer Verlag, 1993, pp. 175–244.

[33] R. Mahnken: Duale Methoden in der Strukturmechanik fur nichtlineare Optimierungsprob-leme. PhD thesis, Institut fur Baumechanik und Numerische Mechanik, Universitat Hanno-ver, 1992.

[34] J.A. Konig: Shakedown of Elastic-Plastic Structures. PWN-Polish Scientific Publishers,1987.

[35] J. Scheer, H. J. Scheibe, D. Kuck: Untersuchung von Tragerschwachungen unter wiederhol-ter Belastung bis in den plastischen Bereich. Bericht Nr. 6099, Institut fur Stahlbau, TUBraunschweig, 1990.

[36] E. Stein, G. Zhang, Y. Huang: Modeling and Computation of Shakedown Problems forNonlinear 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 Kine-matic 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 HardeningMaterial. 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 In-tensity 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 Ap-proaches. 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 Fa-

tigue 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 Ge-

ometry on the Near-Threshold Fatigue Crack Growth in Ferritic Steels. Mat. Sci. Engng.102 (1998) 217–226.


274

12 Parameter Identification for Inelastic ConstitutiveEquations Based on Uniform and Non-UniformStress 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 strainscan be either uniform or non-uniform within the volume. In the second case, the associatedsimulated data are obtained from Finite-Element calculations. A gradient-based optimiza-tion strategy is applied for minimization of a least-squares functional, where the corre-sponding sensitivity analysis if performed in a systematic manner. Numerical examplesfor the uniform case are presented with a material model due to Chaboche with cyclicloading. For the non-uniform case, material parameters are obtained for a multiplicativeplasticity model, where experimental data are determined with a grating method for anaxisymmetric necking problem. In both examples, the effect of different starting valuesand 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 in1991 with the intention to identify material parameters of constitutive models for in-elastic material behaviour. Though the interest for reliable modelling has always beenvery high in the engineering community, up to that time, the concepts for parameteridentification concerning experimental and numerical issues were fairly limited. In par-ticular the state of the art was as follows:

275

* Universitat Hannover, Institut fur Baumechanik und Numerische Mechanik, Appelstraße 9 a,D-30167 Hannover, Germany



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

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

• For optimization of the resulting least-squares functional (at least within the SFB319) evolutionary strategies were preferred.

• In some situations it may occur that more than one set of parameters can give rea-sonable fits. This issue of instability (or even non-uniqueness in the case of identi-cal 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 pa-rameters derived from an optimal least-squares fit of uniaxial experiments do not neces-sarily predict non-uniform deformations. This is due to the facts that (i) a uniaxial ex-periment does not provide enough information to obtain an accurate simulation of thenon-uniform case, and (ii) the ideal test conditions of uniformness often cannot be real-ized in the laboratory. In nearly all mechanical tests deformations eventually cease tobe uniform due to localization, fracture and other failure mechanisms. E.g., non-uni-formness is unavoidable in the case of necking of the sample in tension tests or barrel-ing due to friction of the sample in compression tests. Therefore a main object of pro-ject B8 was to develop a more general approach, which accounts for this inhomogene-ity by performing parameter identification using Finite-Element simulations.

The next issue is concerned with the geometric setting. It is quite obvious thatmaterial parameters obtained from a fit within a geometric linear setting in general donot carry over to the finite-deformation regime. This in particular holds if extremeloads are subjected to the specimen thus yielding large deformations. To this end pa-rameter identification within a geometric non-linear theory has been performed.

In a common – classical – approach, parameter identification is formulated as anoptimization problem, where a least-squares functional is minimized in order to providethe 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 meth-ods) 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 be-cause of its versatility (see e.g. Muller and Hartmann [1] and Kublik and Steck [2]).However, in general, these methods are very time-consuming due to many functionevaluations (several hundred thousand). Thus for reasons of efficiency an optimizationstrategy based on gradient evaluations has been developed.

For determination of the gradient of the associated least-squares functional, basicallytwo variants are known from the literature: (1) The finite-difference method: This tech-

12 Parameter Identification for Inelastic Constitutive Equations

276

nique, though conceptionally very simple in general, is regarded as inefficient due to manyfunction evaluations and accuracy problems. (2) The sensitivity analysis: In this conceptthe gradient is determined analytically consistent with the formulation of the underlyingdirect problem. As part of the work of project B8 the latter concept has been developedfirstly 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 re-sults for the identification process since instability is a typical feature of inverse prob-lems (see Baumeister [3], Banks and Kunisch [4]). To this end two indicators are inves-tigated: We examine the eigenvalues of the Hessian of the least-squares function, andwe study the effect of perturbations of the experimental data on the parameters.Furthermore, for the case of numerically instable results, we introduce a regularizationdue to Tikhonov, which can be interpreted as an enhancement of the basic least-squaresfunctional by adequate model information.

Parameter identification essentially relies on experimental data obtained in the lab-oratory. In this respect it is obvious that for the non-uniform case spatially distributeddata give more information as data evaluated only at certain points, e.g. using straingauges. Therefore optical methods turned out to be the ideal tools in order to obtainthe underlying data sets, and in our examples the experimental data were obtained witha 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, wherethe Finite-Element method is applied. To specify, this work is structured as follows: Inthe next Section 12.2 the basic terminology for identification problems pertaining to thedirect problem and the inverse problem is introduced. In Section 12.3 a systematic con-cept for parameter identification is briefly described for the uniform case, and in Section12.4, it is extended to the non-uniform case. In Section 12.5 two examples for parameteridentification based on experimental data obtained within the Collaborative Research Cen-tre (SFB 319) are presented. In the first example the Chaboche model [8] is consideredwith a sample in cyclic loading, and in the second example we investigate an axisym-metric necking problem of mild steel. Section 12.6 gives a summary of the main resultsof 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 � a (vector-)space with elements � of admissible materialparameters, and g�� u�� is the state equation, which may represent e.g. the (non-lin-ear) state of the discretized form of an initial value problem or the variational form ofan initial boundary value problem. The state equation g may be dependent on the pa-rameters �, both explicitly and implicitly, where the implicit dependency is defined via


277

the state variable �u�� U, and where U is a (function-)space of admissible state vari-ables �u��. With this notation, we formulate the direct problem:

Find u�� U such that g�� u�� 0 for given � � � � �1�

In what follows, we assume existence of the solution u�� Arg �g�� u�� 0� forall � � �.

12.2.2 The inverse problem: the least-squares problem

Let �D denote an observation space, and let �d � �D denote given data from experiments.In general experimental data are not complete, e.g. for cyclic loading tests very oftenthey are available only for a part of the cycles. To account for this possible incomplete-ness, we introduce an observation operator � mapping the trajectory u�� to points� u�� in the observation space (Banks and Kunisch [4], p. 54). With this notationwe formulate the inverse problem:

Find � � � such that � u�� d for given �d � �D � �2�

An identification process based on experimental data is typically influenced by twotypes of errors: Using the notations �u for the true state and � for the correct param-eter vector, then the following situations may arise (Banks and Kunisch [4]):

• � �u �� d due to measurement errors,• �u �� u�� due to model errors.

In general, the first error type is taken into account by statistical investigations of thedata. The second type is reduced by increasing the complexity of the model, which ingeneral is accompanied by an increase of material parameters np.

Referring to the classical definitions of Hadamard [9], a problem is well-posed if theconditions of (i) existence, (ii) uniqueness and (iii) continuous dependence on the data forits solution are satisfied simultaneously. If one of these conditions is violated, then theproblem is termed ill-posed. Since in practice the number of experimental data is largerthan the number of unknown parameters, problem (2) in general is overdetermined, thusexcluding the existence of a solution due to measurement and/or model errors. The clas-sical strategy therefore uses an optimal approach of simulated data u�� and experimentaldata �d, thus replacing problem (2) by the least-squares optimization problem:

Find � � � such that for given �d � �D � f �� 12�� u�� d�2

�D � min��

� �3�

• Remarks

1. In practice, experimental data are given at discrete time- or load-steps. Therefore,for the following discussions it is natural to set �D � IRndat for the observation space,


278

with ndat as the number of experimental data. Furthermore, very often parameters areindependent of each other such that � � IRnp can be separated, where np is the numberof material parameters, Next, we use the short hand notation � u�� d�� 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:

f �� 12�d�� d�2

2 � min��

� � ��np

i�1

�i � �i �� ai � �i � bi� � �4�

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 nu-merically instable solutions, i.e. small variations of �d then lead to large variations ofthe parameters �. These difficulties are caused if

(a) the material model has (too many) parameters, which yield (almost) linearly de-pendencies within the model, or if

(b) the experiment is inadequate in the sense that some effects intended by the modelare not properly “activated”.

It has already been mentioned that a typical step for reducing the model error is to in-crease the complexity of the model, which generally is accompanied by an increase ofthe material parameters. A typical example is the modification of the standard J2-flowtheory 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 Equa-tion (8) in the forthcoming Section 12.3) in order to account for non-linear kinematichardening effects. In doing so, it should be realized that the introduction of additionalmaterial parameters may also result into the aforementioned numerical instability forthe identification process if appropriate steps are not performed when planning the ex-periment. To summarize, the contradictory requirements for numerical stable resultsand reducing the model error have to be carefully balanced.3. In some cases, even non-uniqueness for the parameter set may occur: This was ob-served by Mahnken and Stein [10, 11] for two material models under certain loadingconditions. The main consequences are that at least cyclic loading becomes necessaryin case of the Chaboche model [8], and for identification of the Steck model [12] ex-periments have to be performed at different temperatures.4. As a consequence of the above Remark 2, it is strongly recommended to study theeffect of perturbations of the experimental data on the parameters. This may indicatepossible instabilities of the identification process. Furthermore, the eigenvalue structureof the Hessian of f �� gives further information about the stability. However, a system-atic strategy to detect possible instabilities so far is not available.5. A mathematical tool, suitable to overcome possible numerical instabilities, is a regu-larization of the functional in Equation (4), and this leads to the more general problem:

f�� 12

��W��d�� d��

2

2

� �

2

��W��

2

2

� min��IRnp

� �5�


279

Here, the matrices W� � IRndat � IRndat and W� � IRnp � IRnp , the scalar � � IR� and thea priori parameters �� IRm are regularization parameters (see Baumeister [3]). Notethat the first part of the functional in Equation (5) is also obtained when consideringparameter identification based on statistical investigations in the context of a Maxi-mum-Likelihood method in order to account for measurement errors. It is noteworthythat 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 in-terpretation of some parameters, obtained e.g. by “hand fitting”, into the optimizationprocess if numerical instabilities occur. However, a systematic concept for determina-tion of the regularization parameters in the context of parameter identification for visco-plastic material models so far is not available.

6. Problems of the above kind like Equations (4) or (5) with separable constraintsmay be solved with the projection algorithm due to Bertsekas [15]:

��j�1� ��j� ��j� �H�j��f ��j�� i �� min�bi� �max��i� ai�� i � 1� � � � � np � �6�

Note that the above iteration scheme requires the gradient of the associated least-squares functional. This task is generally performed in the sensitivity analysis, wherethe gradient is determined consistent with the formulation of the underlying direct prob-lem. Note also that the iteration matrix �H has to be “diagonalized” in order to insuredescent properties of the search directions for algorithm (see Bertsekas [15] and Mahn-ken [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 forthe uniform case within a geometric linear setting. A detailed description is given inMahnken and Stein [11, 17].

12.3.1 Mathematical modelling of uniaxial visco-plastic problems

Let � � �0� T� be the time interval of interest. The uniaxial stress is designated by� � �11 � � � IR, while �el and �in � � � IR, are the elastic and inelastic parts of thesmall strain tensor components �in

ij and �elij , respectively. The model equations represent-

ing one-dimensional visco-plasticity with small strains are summarized as follows:

� � �el � �in additive split of total strains � �7 a�

�el � 1E� elastic strains � �7 b�


280

��in � ��in�� q� �� in� � � � � �� evolution for inelastic strains � �7 c�

�q � ��q�� q� �� in� � � � � �� evolution for internal variables . �7 d�

Here, additionally, we defined the temperature �, the elastic modulus E, and � � IRnp isa vector of np material parameters characterizing the inelastic material behaviour.

There exists a great variety of constitutive relations in the literature according tothe above skeletal structure (Equations (7a) to (7d)) (see e.g. Miller [18], Lemaitre andChaboche [19] and references therein). Many approaches intend to provide for a num-ber of different characteristic effects such as strain rate-dependent plastic flow, creep orstress relaxation. In doing so, a yield criterion with the inherent specification of loadingand unloading conditions as in time-independent classical plasticity is not needed. Theresulting equations are currently referred to as “unified models”. Concerning the inter-nal variables, in principle they are argued for macroscopic or microscopic reasons de-pending 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 ofChaboche [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

��in �FK �

� �n�

sign �� if F � 0

0 else

�� 8 a�

�R � b�q R��in sign ��in� isotropic hardening �8 b�

�� c�� sign ��in��in kinematic hardening �8 c�

F � �� sign �� R k� overstress �8 d�

shall be considered, where � �� n��K �� k�� b� q� c� ��T is the vector of material parametersrelated to the inelastic material behaviour.

In addition to the Equations (7a) to (7d), we assume that initial conditions

��t � 0� � �0 � �in�t � 0� � �in

0 � q�t � 0� � 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 complemen-tary tests, i.e. strain-controlled experiments, where experimental data are given for astress distribution ��t�� t � � .

12.3 Parameter Identification for the Uniform Case

281

12.3.2 Numerical solution of the direct problem

We define N as the number of time steps �tk � tk tk 1� k � 1� � � � �N� t0 � 0� tN � T .Using the second order midpoint-rule at each time step, from Equations (7a) to (7d),we obtain the update relations

�k � �k 1 � �tk ��ink 1�2 � ��el

k 1 � �10�

qk � qk 1 � �tk �qk 1�2 � �11�

where we applied the notation:

��ink 1�2 � ��in�1�2��k 1 � �k� � 1�2�qk 1 � qk�� 12�

�qk 1�2 � ��q�1�2��k 1 � �k� � 1�2�qk 1 � qk�� 13�

��elk 1 � 1

E��k �k 1� � �14�

Since the state variables �k and qk are not known in advance, the following non-linearsystem of equations has to be solved at each time step:

gk�1��k� qk� �� k �k 1 �tk ��ink 1�2 ��el

k 1 � 0 � �15�

gk�2��k� qk� �� qk qk 1 �tk �qk 1�2 � 0 � �16�

Defining Gk �� gk�1� gTk�2�T and a vector of state variables Yk �� k� qT

k �T, Equations(15) and (16) may be summarized as:

Gk�Yk� � 0 � �17�Referring to the notation of Section 12.2.1, Equation (17) will be termed as the stateequation, describing the state of the variables Yk �� k� qT

k �T at the k-th time step.Furthermore, using the notation of Section 12.2.1, we have the direct problem:

Find Yk�� such that Gk�Yk�� 0� k � 1� � � � �N for given � � � � �18�The iterative solution of Equation (17) is obtained with a Newton method. Details ofthis strategy with applications to the material models of Bodner-Partom, Chaboche andSteck 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 timeintegration �tk�Nk�1 � � and for the experimental data �texp

k �ndatk�1 � � do coincide for both


282

the numerical values �k�� tk� �� and the observations ��k � ��tk�� k � 1� � � � �N. Thefollowing considerations can be extended to more complex situations in a straightforwardmanner. The resultinginverseproblemthenbecomes the least-squaresoptimizationproblem:

Find � � � such that for given �d � �D �

f �� 12

Nk�1

��k�� k�2 � min��IRnp

�

� ��np

i�1

�i � �i �� ai � �i � bi� � �19�

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 materialparameters and an inner loop for iteration of the state variables �Yk�� are performed. In theouter loop the Bertsekas algorithm(Equation (6)) is applied,where the gradient is determinedin a sensitivityanalysis. For detailspertaining to this strategy with applications to the materialmodels 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 stressand strain distributions during the experiment cannot be guaranteed due to the experi-


283

Figure 12.1: Schematic flow chart of the optimization strategy for the uniform case with outerand inner iteration loops.

mental conditions or failure mechanisms. Therefore, we will consider parameter identi-fication with Finite-Element simulations in order to take into account inhomogeneitieswithin the sample. In what follows, we give a brief review for a geometrically non-lin-ear continuum-based formulation of the direct problem as a variational problem andfurthermore of the associate least-squares problem. For solution of these problems, astandard linearization procedure is applied for the direct problem and a sensitivity anal-ysis is performed for the inverse problem. In the forthcoming sections, we will com-ment on the similarities of these associated concepts. More details of our approach aredocumented in Mahnken and Stein [10, 21] for the geometric linear case, and in Mahn-ken 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 smoothboundary ��, and let X � � � IRndim be the position vector in the Euclidian space IRndim

with spatial dimension ndim � 1� 2� 3. We shall denote by �� and �� those parts ofthe boundary ��, where configurations are prescribed as �� and boundary tractions areprescribed as �t, respectively. As usual, we assume �� and�� . In addition, �b denotes the body force per unit volume. As before, wedefine �� t0�T � � IR� as a time interval of interest, and � � IRnp designates the(vector-)space of material parameters.

Following Barthold [23] the fundamental mapping for describing the current con-figuration of the body for varying time t � � and varying parameter � � � is given as:

�� IRndim ,�X� t� �� X� t� ��

�20�

As usual, we restrict ourselves to configurations � satisfying J �� det�F� � 0 and� � �� on ��, where we use the shorthand notation F �� F�X� t� �� X� for the de-formation gradient at �X� t� ��. The exposition that follows crucially depends on the ba-sic assumption:

X � ��X� t � t0� �� X� �� 21�

i.e. the initial configuration � at time t � t0 is independent of the parameter set �. It isnoteworthy that this restriction, e.g., does not hold for more complex situations in shapeoptimization, and would necessitate the introduction of a reference configuration invariantof the design variables � (Barthold [23], Haber [24]). Thus, we will regard the set�X� t� �� as the independent variables in the ensuing considerations.

An illustration of the mapping (Equation (20)) for the body � at fixed time t fortwo different parameter sets �1� �2 � � is shown in Figure 12.2.


284

12.4.2 The direct problem: Galerkin weak form

Let � � � be given, and let us assume a partition of the time interval � � �Nk�1

�tk 1� tk�into N subintervals (for problems of elasticity and plasticity tk refers to the load step).Denoting by �k � �� tk� �� the configuration at time tk for the parameter set �, thebalance equation of linear momentum and the set of Dirichlet and Neumann boundaryconditions at the (k)-th step read:

div�k � �b � 0 in � �

�k � ��k on ��

�kn � �tk on �� 22�

Here, �k designates the Cauchy stress tensor. Using the notation �� for the L2 dualpairing on � of functions, vectors or tensor fields, an equivalent formulation is the clas-sical weak form (principal of virtual work) of the momentum equations at time tk. In aspatial description this results into the direct problem:

Find �k such that g��k� � �� d� !t�tk �g!tk � 0 � �u and for given � � � � �23�

where the Kirchhoff stress tensor � � J� is introduced. Furthermore, a spatial rate of de-formation tensor induced by the virtual displacement �u is defined as d� �� sym��u�,and �g �� b " �u � ��t " �u �� designates the external part of the weak form. For the caseof inelastic problems the above set of equations has to be supplemented by initial condi-tions Z�X� t0� �� Z0, where Z denotes a set of history variables.

The iterative solution of the non-linear problem (Equation (23)) is based on a stan-dard Newton method, in which a sequence of linearizations of the weak form (Equation(23)) is performed. To this end the Gateaux derivative ��g��k� of problem (Equation (23))as shown in Table 12.1 is determined. Here, l� �� u and d� �� sym�l�� are a velocitygradient and a spatial rate of deformation tensor, respectively, induced by the linearizationincrement �u. Additionally, c is the fourth order spatial material operator.


285

Figure 12.2: Illustration of two-parameter-dependent configurations at fixed time.

12.4.3 The inverse problem: constrained least-squares optimization problem

As in Section 12.3, we assume identical time (load) steps �tk�Nk�1 and observationstates �tj�ntdat

j�1 , where experimental data �dj � �D are available, and where �D denotes theobservation space. In particular, �dj may contain stresses, strains, displacements, reactionforce fields etc. Since in general only incomplete data are available from the experi-ment, we introduce an observation operator � mapping the configuration trajectory�k � �� tk� �� k � 1� � � � �N to points of the observation space �D. Note that this defini-tion of � 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 �k. Thenwe consider the least-squares optimization problem:

Find � � � such that for given �d � �D � f �� 12

Nk�1

��k �dk!!2�D � min��

� �24�

where �k 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 withthe Bertsekas algorithm (Equation (6)). Then determination of ��k, needed for evalua-tion of the least-squares functional (Equation (24)), is performed only at the convergedstate of the direct problem, i.e. when problem (Equation (23)) is satisfied at the k-thtime (load) step.

Next we will briefly resort to the parameter sensitivity ��k. To this end firstly,the parameter sensitivity ��g��k� of the weak form (Equation (23)) is determined, atequilibrium, analogously as in the linearization procedure. The resulting expression for��g��k� is given in Table 12.1 in a spatial setting, where now a spatial rate of deforma-tion tensor induced by the parameter sensitivity ��k � �� is defined asd� �� sym ��. Note that ��g��k� also requires the spatial material operator c of thelinearization procedure. Furthermore we defined the sensitivity-load term ��p

�� d� !t�tk ,which excludes dependencies of � via the configuration �. The determination of thisterm becomes a major task of the sensitivity analysis, and we refer to Barthold [23]and Mahnken and Stein [22] for further details.


286

Table 12.1 : Weak form, Gateaux derivative for linearization and linear equation for parametersensitivity in a spatial formulation.

• Weak form (principle of virtual work)�� d� !��

��!��

• Gateaux derivative for linearization�� c � d�� d� � l�� d� !��

• Linear equation for parameter sensitivity�� c � d�� d� � l�� d� � �� d� !��

� �

In the practical implementation, firstly, the sensitivity load term is determined in apreprocessing procedure consistent with the underlying integration algorithm. Havingobtained ��k by solution of the associate linear equation, the total derivative ��wk��of any quantity w��k� is performed in a postprocessing procedure. Further details ofthe procedure are described in Mahnken and Stein [22].

We close this section with the remark that the above results can be easily ex-tended 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 deter-mined in the case of an aluminium/magnesium alloy. The underlying experimental datawere obtained from project A2 (Prof. Lange [26]).

The experiments were performed at room temperature for a cylindrical hollowspecimen with an outer radius of 28 mm and a thickness of 2 mm. The specimen weresubjected to a periodic strain of an amplitude of �max � 0�3% at a strain rate of�� 0�2% s–1. 110 cycles were generated during the test, however, experimental dataare available only for 20 cycles out of these. Youngs modulus has been predeterminedas E � 1�09 " 105 MPa.

As an objective function for the inverse problem the simple least-squares function

12.5 Examples

287

Figure 12.3: Schematic flow chart of the identification process for the non-uniform case withouter and inner iteration loops.

f �� 12�� 2

2 �25�

is minimized, which is the analogue of Equation (19) for the case of strain-controlled ex-periments. Note that due to the incompleteness of the data set �� contains data only for 20cycles out of the total of 110. The minimization was performed with an evolutionary strat-egy as described in Schwefel [27] and with the Bertsekas algorithm (Equation (6)). For thefirst method we used three “parents” and “twenty descendants”, whilst for the latter theBFGS-matrix was used as an iteration matrix and a Gauss-Newton matrix for precondi-tioning. The computer runs were performed on an IBM-250T.

The starting vector and the solution vectors are given in Table 12.2. Concerning theBertsekas algorithm, three different runs were made. Run 1 and Run 2 were started withthe vector in the second column, however, for Run 2, a regularization was performedusing the extended functional Equation (5). Here, for B� and for B�, the unity matrix ischosen, and we set � � 10 5. It can be seen that no convergence was attained for Run1 after 2000 iteration steps, whilst minimization with the regularized functional attainedconvergence after 201 steps. The corresponding minimal eigenvalue of the Hessian atthe solution point is 7.62 · 10–2, thus indicating stable results. This also is confirmed byRun 3, where each data was perturbed stochasticly with a maximal value of 10%, andwhere the effect of this perturbation is negligible. In the last two columns of Table 12.2results for the evolutionary strategy are shown. After 897 iterations the results are stillpoor (Run 4), and after 10256 iterations and 168 h, the value for the objective functionis still above that obtained by the Bertsekas algorithm (Run 2) in 24 min.


288

Table 12.2: Cyclic loading for AlMg: starting and obtained values for the material parameters ofthe Chaboche model for AlMg in case of different optimization strategies and least-squares func-tions. Concerning Run 3 see Section 12.5.1. ITE and NFUNC denote the number of iterations andfunction evaluations, respectively.

Bertsekas algorithm Evolutionary strategy

Start Run 1 Run 2 Run 3 Run 4 Run 5

�� [–] 5.0 · 100 4.582 · 100 1.360 · 101 1.360 · 101 4.971 · 100 1.250 · 101

� � [MPa] 1.0 · 102 2.344 · 102 4.242 · 101 4.242 · 101 1.972 · 102 3.896 · 101

�� [–] 1.0 · 102 5.195 · 100 4.824 · 100 4.824 · 100 5.115 · 100 5.068 · 100

� [MPa] 1.0 · 102 6.233 · 101 6.827 · 101 6.827 · 101 6.488 · 101 6.697 · 101

� [–] 1.0 · 102 0.206 · 10–1 1.542 · 103 1.542 · 103 1.173 · 102 1.546 · 103

� [MPa] 1.0 · 102 9.840 · 104 4.719 · 101 4.719 · 101 1.792 · 102 4.736 · 101

�� [MPa] 1.0 · 101 0.000 · 100 0.000 · 100 0.000 · 100 8.543 · 10–1 2.238 · 100

�� 3.491 · 105 9.620 · 103 2.135 · 103 3.335 · 103 8.936 · 103 2.135 · 103

�� – – 3.582 · 10–5 – – –CPU [min] – 192 24 – 1440 605 ·103

ITENFUNC

– 20002072

201250

––

89717752

10 25620 493

Remark – no conver-gence

regularized perturbed – –

12.5 Examples

289

Figure 12.4: Cyclic loading for AlMg. Up: Stress versus time for the solution parameter set forthe first 18 out of 110 cycles. Note the incompleteness of the experimental data set. Down: Stres-ses versus strains for three different cycles. The numbers 1, 30, 110 correspond to the specific cy-cles.

� ��

Figure 12.4 depicts the stresses versus strains for three different cycles for the so-lution vector of the material parameters. It can be seen that very substantial agreementof experimental and simulated data is obtained, except for the first cycle, where themodel is not able to simulate the horizontal plateau. This explains the relatively highmodel 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 ma-terial of the specimen is a mild steel, Baustahl St52, due to the german industrial codesfor construction steel. The experimental data were obtained with a grating method. Forthis purpose, firstly, grid marks were positioned on the surface of the sample, and thesewere recorded by a digital CCD-camera at different observation states ti� i � 1� � � � ntdat

in the displacement-controlled experiment. In Figure 12.5, the sample with the gratingis shown at four observation states 5, 7, 10, 13 as introduced in Figure 12.7. More de-tails 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 methodsin order to obtain the final data for the identification process. This task is described inmore detail in the contribution of project C1 (Dr. Andresen [5]).

The elastic constants are E � 20 600 kN/cm2 for Youngs modulus and � 0�3 forPoissons ratio. The material is assumed to be elasto-plastic, modelled by large strain multi-plicative von Mises elasto-plasticity with non-linear isotropic hardening summarized inTable 12.3 (see Simo and Miehe [28] for further details). Solution of the direct problemis done with a product-formula algorithm according to Simo [29]. The solution of the in-verse problem is based on the general setting described in the previous section. Details


290

Figure 12.5: Axisymmetric necking problem: photographs with a CCD-camera of the sample andthe grating at four different observation states NLST as introduced in Figure 12.7.

12.5 Examples

291

Table 12.3: Large strain multiplicative von Mises elasto-plasticity.

� � �� b�� g � Kirchhoff stress

��

� �� yield function

� �� flow stress

�

#��b��b��

�� flow rule

��

��

variable evolution

� $ �� loading and unloading conditions

� �� vector of material parameters

Table 12.4: Axisymmetric necking problem: starting and obtained values for the material parame-ters of a mild steel, Baustahl St52, for three different optimization runs. � �� denotes the numberof iterations.

Run 1 (Q1/E4) Run 2 (Q1/E4) Run 3 (Q1/E4)

Starting Solution Starting Solution Starting Solution

�� [MPa] 300.0 360.26 400.0 360.26 400.0 346.09� [–] 10.0 3.949 20.0 3.949 20.0 3.951� [MPa] 800.0 436.72 2000.0 436.72 2000.0 419.73

�� [–] 1.738 · 104 4.210 · 102 3.012 · 104 4.210 · 102 2.685 · 104 3.245 · 102

Perturbed no no yes� �� 34 34 39

Figure 12.6: Axisymmetric necking problem: different levels for spatial discretization.

concerning the sensitivity analysis consistent with the product-formula algorithm of Simo[29] for determination of the load term �p

��k can be found in Mahnken and Stein [22].An axisymmetric enhanced strain element (Q1/E4) described by Simo and Ar-

mero [25] is used in the element formulation, and only a quarter of the bar is consid-ered for discretization using the appropriate symmetry boundary conditions.

The object is to identify the 3 parameters b� q� �0 of Table 12.3, which character-ize the inelastic behaviour of the material. To this end the following least-squares func-tional is minimized


292

Figure 12.7: Axisymmetric necking problem: comparison of simulation and experiment. Up: Loadversus elongation. Down: Necking displacement versus total elongation.

f �� ntdat

i�1

12��

j��

j�2 �

ntdat

i�1

12�w�Fi�� Fi��2 � �26�

where ��j

and �Fi��, �Fi, i � 1� � � � � ntdat denote data for configurations and the total loads,respectively. The number of load steps is N � 40, the number of observation states isntdat � 9, and we have nmp � 12 for the number of observation points. A multilevelstrategy is applied in order to accelerate the optimization process by using solutions oncoarser grids as starting values on finer grid. In this example the total number of levelsis five (Figure 12.6).

In Table 12.4 results for the parameters � of three different runs are listed. WhilstRun 1 and Run 2 differ in their starting vectors in order to take into account possiblelocal minima, the purpose of Run 3 is to investigate the effect of a perturbation of theexperimental 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 thatRun 1 and Run 2 give identical results, thus indicating no further local minima. The re-sults for Run 3 differ only slightly from the results of Run 1 and Run 2, thus indicat-ing a stable solution with respect to measurement errors.

In Figure 12.7 the results for the total load versus total elongation and maximalnecking displacement versus total elongation are compared for simulation and experi-ment. Figure 12.8 depicts a 2-D illustration of the final Finite-Element grid at differentobservation states and the corresponding experimental data along the side of the sam-ple. It can be observed that for both quantities of different type, displacements andforces, excellent agreement is obtained after optimization.

12.5 Examples

293

Figure 12.8: Axisymmetric necking problem: comparison of experiment and Finite-Element simu-lation for the configurations at four different observation states NLST as introduced in Figure12.7. The circles represent the experimental data.

12.6 Summary and Concluding Remarks

In this contribution the main issues of project B8 of the research period 1991–1996 foridentification of parameters for inelastic constitutive equations were presented. Themain purpose of the project was to develop a general concept of gradient-based optimi-zation 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 stressesand strains within the sample during the experiment. The main results of the projectand the cooperation with other projects from the Collaborative Research Centre (SFB319) are listed as follows:

Theoretical results

• A gradient-based optimization algorithm has been developed for minimization of aleast-squares functional. In particular a projection algorithm due to Bertsekas is ap-plied, which accounts for possible upper and lower bounds for the parameters. Qua-si-Newton methods with Gauss-Newton preconditioning are used as iteration ma-trices.

• A unified strategy for an analytical sensitivity analysis in the case of uniform stressand strain distributions is obtained, valid for a certain class of constitutive equationswith internal variables. The resulting scheme is consistent with the correspondingtime integration scheme, and as a main result a recursion formula is obtained. Ithas 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. Asfor the uniform case it is consistent with the time-integration scheme.

• Investigations of uniqueness (or stability, respectively) for the inverse problem forcertain material models were performed (see Remark 3 of Section 12.2.2). Themain consequences are that at least cyclic loading becomes necessary in case of theChaboche model [11], and for identification of the Steck model experiments have tobe performed at different temperatures [10].

• A regularization technique for stabilization of the least-squares problem based on apriori information has been introduced. Applications were done for the Bodner-Par-tom model [11].

Numerical results

• Parameter identification for the methods of Chaboche, Steck and Bodner-Partombased on experimental data for experiment with uniform stresses and strains wasperformed.


294

• Comparative results with the evolutionary strategy showed great advantage of gradi-ent-based schemes with respect to execution time.

• Parameter identification with the Finite-Element method was performed in theframe of non-linear multiplicative plasticity using experimental data obtained with agrating 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 prob-lem 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 inDer Bauingenieur (see Andresen et al. [30]).

• For parameter identification for an aluminium/magnesium alloy subjected to cyclicloads, data of project A2 (Prof. Lange [26]) were used.

Concluding remarks

The concepts proposed in this paper provide a flexible approach for identification of in-elastic material models. This opens the possibility to obtain more insight into the non-uniformness of the samples during the experiment and on the reliability of the numeri-cal results. However, some open questions remain to be considered for future work:

• Further development in this area should take into account phenomena such as dam-age, localization and temperature-dependent effects, which very often are highlynon-uniform during the experiment.

• In practice we know that measurement techniques possess limited accuracy. A prob-abilistic investigation of these dispersion phenomena can be done with the Maxi-mum-Likelihood method (see Bard [13], Pugachew [14] and project B1 (Prof. Steck[31]). Furthermore we know that repetition of the same experiment with differentsamples in general yield different values. Reasons for this scattering of the datamay be due to inhomogeneities, load uncertainty, production, nature of the phenom-enon. Therefore this randomness also necessitates a probabilistic approach, wheree.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 whenperforming parameter identification in the future.

• In the work done so far the effect of discretization errors both in space and in timeis not considered, especially in the frame of a proper error control. Therefore itwould be of interest to take into account adaptive strategies in the context of theoptimization process.

12.6 Summary and Concluding Remarks

295

• The shape of the least-squares functional, which is the objective function of the re-sulting optimization problem, may not be convex, and thus different local minimamay 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 suchthat an error-controlled adaptive modelling might be necessary.

References

[1] D. Muller, G. Hartmann: Identification of material parameters for inelastic constitutivemodels 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 MultiaxialExperiments. In: D. Besdo, E. Stein (Eds.): IUTAM Symposium Hannover 1991, Finite In-elastic 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. Birk-

hauser, 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 Op-

tical Measuring Methods. This book (Chapter 13).[7] U. Peil, J. Scheer, H.-J. Scheibe, M. Reininghaus, D. Kuck, S. Dannemeyer: On the Behav-

iour 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 An-

isotropic 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-Ele-

ment-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 Analyti-cal 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. ofPlast. 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 con-

straints. SIAM J. Con. Opt. 20(2) (1982) 221–246.[16] R. Mahnken: Duale Verfahren fur nichtlineare Optimierungsprobleme in der Strukturmecha-

nik. Dissertation, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Uni-versitat Hannover, F 92/3, 1992.


296

[17] R. Mahnken, E. Stein: Gradient-Based Methods for Parameter Identification of ViscoplasticMaterials. 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 AppliedScience, 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 ma-terials. Trans. ASME, J. Appl. Mech. 42 (1975) 385–389.

[21] R. Mahnken, E. Stein: A Unified Approach for Parameter Identification of InelasticMaterial 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 inPrinicpal Directions. Comp. Meths. Appl. Mech. Eng. 147 (1997) 17–39.

[23] F. J.B. Barthold: Theorie und Numerik zur Berechnung und Optimierung von Strukturen ausisotropen, hyperelastischen Materialien. Dissertation, Forschungs- und Seminarberichte ausdem Bereich der Mechanik der Universitat Hannover, F 93/2, 1993.

[24] R. B. Haber: Application of the Eulerian Lagrangian Kinematic Description to StructuralShape Optimization. Proc. of NATO Advanced Study Institute on Computer-Aided OptimalDesign, 1986, pp. 297–307.

[25] J.C. Simo, F. Armero: Geometrically Nonlinear Enhanced Strain Mixed Method and theMethod of Compatible Modes. Int. J. Num. Meth. Eng. 33 (1992) 1413–1449.


[27] K.P. Schwefel: Numerische Optimierung von Computer-Modellen mittels der Evolutions-strategie. Birkhauser 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 theClassical 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: Parameteridentifi-kation fur ein plastisches Stoffgesetz mit FE-Methoden und Rasterverfahren. Der Bauingeni-eur 71 (1996) 21–31.

[31] E. Steck, F. Thielecke, M. Lewerenz: Development and Application of Constitutive Modelsfor the Plasticity of Metals. This book (Chapter 4).

References

297

13 Experimental Determination of Deformation-and 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 de-scribe the inelastic behaviour of metallic materials. It is needed first of all to observesuch a behaviour in order to get knowledge of the process of the corresponding meth-ods. Furthermore, the parameters for such material laws must be measured. They canbe achieved from experimentally determined, multi-dimensional load-deformation distri-butions. The experiment is finally necessary for the comparison with the calculation inorder to verify the implanted material laws. The requirements of the measuring meth-ods 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 totwo- or three-dimensional distributions of the searched values, also such measuringmethods are needed, which allow larger object areas to be analysed connected and two-dimensional. As one must furthermore plan on a large local change of the material be-haviour, a high local resolution of the measuring method is also required. In addition,measuring systems must be developed, with which deformation- and strain fields canbe measured even in the transitional areas from inelastic to elastic material behaviour.

The measurements should preferably be able to be executed on the original be-cause the model laws for the transfer of results from the model to the original are ingeneral very complicated, especially for inelastic material behaviour. Inelastic materialbehaviour is often observed at high temperatures up to approx. 1000 �C. The measuringmethods should also be usable in such temperature areas. A measurement directly onthe testing machine in the testing field during an ongoing test is practical.

298

* Technische Universitat Braunschweig, Institut fur Messtechnik und Experimentelle Mechanik,Schleinitzstraße 20, D-38106 Braunschweig, Germany



Finally, measuring methods without contact and interaction are advantageous. Allof these requirements lead to the development and the use of the optical field-measur-ing techniques. The corresponding methods are distinguished by the following charac-teristics.

13.3 Characteristics of the Optical Field-Measuring Methods

First, two-dimensional structured patterns are generated in the form of intensity distri-butions, which are related to the searched values of the considered object surface. TheFigures 13.1 and 13.2 show two such patterns. In Figure 13.1, there are two groups oflines, which consist of straight lines of constant width. They include an angle of 90 �and form the so-called cross grating. If this is e.g. firmly attached to the consideredsurface, 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 sur-face. The remitted beams interfere. As a result of the distribution of different ampli-tudes and phases, a granular-like intensity distribution comes into existence, which iscalled the Speckle effect [1].

13.3 Characteristics of the Optical Field-Measuring Methods

299

Figure 13.1: Cross grid.

Figure 13.2: Speckle pattern.

In a second step, such patterns are recorded in the image plane by recording cam-eras, and their image points are determined from the digital image processing. Fromthis, 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 meth-od, 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 ob-ject surface. This is recorded from two or more different positions and orientations inreference to the object by cameras [4] (Figure 13.3). By retransforming the digitally de-termined 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 deformationof the object leads to the deformation vector. The change of the distance between twoneighbouring object points related to their original distance describes the strain [5].

The stereophotographic set-up of Figure 13.3 represents the simplest arrangementaccording to the photogrammetric principle [6]. There in general, only the local vectorof an object point is of interest.

In case of plane deformation of the object, only one recording camera of the opti-cal set-up is needed. With regard to the previously described 3-D object-gratingmethod, then this is called a 2-D object-grating method.

The following results have been achieved for the development of this measuringmethod for the deformation analysis of objects with inelastic material behaviour.

13 Experimental Determination of Deformation- and Strain Fields

300

Figure 13.3: Principle of the 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 ob-ject surface, but which also can be recognized at high temperatures. The well-knownscreen print principle was the basis for the further development [7].

If a mixture of TiO2-particles of approx. 0.3 �m diameter and ethanol is sprayedthrough a screen-like mask on the polished object surface, then a grating-like structurecomes into existence after the ethanol is vaporized and the mask is removed. This iscomposed of the blank polished object surface and local limited fields, which each con-sist of a great number of such TiO2-particles. Figure 13.4 shows a REM-picture ofsuch a grating structure.

By dark field illumination, the object surface appears dark, and each grating fieldis light due to the diffuse remission of the individual TiO2-particles. Figure 13.5 is oneexample for this type of illumination and also the recognizability of the grating at hightemperatures [8].


301

Figure 13.4: TiO2-grating in REM.

Figure 13.5: Cross grating at 850 �C.

This type of grating structure can also follow a very large deformation of the ob-ject 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 tempera-tures. The specimen with the attached TiO2-grating is enclosed by a radiation heater inthe testing machine. It is heated up by infra-red radiation. The visible radiation partserves as a dark field illumination of the test surface with the grating. This is recordedthrough 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. Inthe case of the 3-D object-grating method, the heater would have to be furnished witha second window. However, for this, another heater or illumination concept would be


302

a) b)Figure 13.6: Cross grating on a curved object surface; a) not deformed; b) strongly deformed.

Figure 13.7: Principle of the optical deformation analysis at high temperatures (top view).

necessary in order to produce the necessary dark field illumination of the grating forboth cameras also at high temperatures. In addition, the test set-up cannot be calibratedat 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 determin-ing the searched deformation are perhaps superimposed by unwanted rigid body move-ments of the considered specimen. These can lead to large errors in determining the de-formation according to the 2-D grating method. In Figure 13.8, the possible grating dis-tortions 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 imagecoordinates distorted by rigid body movements to undistorted ones:


303

Figure 13.8: Grating distortions as a result of translatory and rotatory rigid body movements in re-lation to the camera coordinate system.

xt � a1 � a2x � a3y� a4xy � �1�yt � a5 � a6x � a7y� a8xy � �2�

The following steps are necessary in order to compensate possible virtual deformationswith an otherwise in-plane deformation of the object:

A reference object with an attached grating is fastened to the specimen so that therigid body movements of both are equal, but such that the reference object is not de-formed.

After orienting the plane specimen surface and reference object parallel to the im-age plane of the recording camera, the simultaneous recording of the specimen and ref-erence grating takes place.

From the coordinates of the reference grating referring to the non-deformed and de-formed specimen state, the parameters for the retransformation i.e. its virtual deformationcan be determined from Equations (1) and (2). With the aid of the now known retransfor-mation instruction, the true deformation of the specimen can be obtained from the imagecoordinates referring to both loading states observed in the specimen. Figure 13.9 showstwo fields of lines of the same strain, one with (a) and the other without (b) virtual strains.


304

Figure 13.9: Lines of the same strain a) with and b) without virtual strains.

13.4.5 3-D deformation measuring

A strategy has been developed for the exact calibration of the system for a 3-D defor-mation measurement [10, 11]. For this purpose, formulations and algorithms have beendeveloped, which are based on the photogrammetric principle. Afterwards, the innerand outer orientation of the cameras used are determined with an appropriate calibra-tion 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 whenusing a camera with 1024 ×1024 pixels has been put together. These refer in one caseto 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 mea-suring area or at the same resolution to recording a larger object area.

This method can currently be used for temperatures up to 1000 �C and measuringareas from 0.1 ×0.1 mm2 to any size. It primarily provides the field of the local vectorsof the observed object points and the displacement- and strain fields derived from them.

13.5 Speckle Interferometry

13.5.1 General

When developing material laws for the inelastic behaviour of metallic materials, thetransition to elastic procedures must be included. Therefore, optical field-measuringmethods are needed, with which both areas can be analysed.

Since the object-grating method despite of all of its advantages is not sensitiveenough for determining strains, which are smaller than 0.3%, a supplementary measur-ing method had to be developed, with which lower scales are also attainable.


305

Table 13.1: Example for the accuracy of the object-grating method.

Camera Object

Measuring area: 1024 ×1024 Pixel 20 ×20 mm2

Number of measuring points approx.: 75 ×75 75 ×75Accuracy of the displacement approx.: 0.02 Pixel 0.4 �mReference length of the strain: 13 Pixel 0.25 mmAccuracy of the strain approx.: 0.2% 0.2%

The measuring method developed for this is based on well-known optical paths ofrays of the Speckle interferometry for measuring the in- and out-of-plane displacementof an object surface [12, 13].

With this, one can e.g. make each one of the both in-plane components of thedisplacement visible directly in the form of correlation fringes without the out-of-planecomponent being included (Figure 13.10). The visibility and thus also the possibility ofobtaining qualitative values on-line can be done e.g. on a video monitor. This offersthe possibility of selectively controlling the deformation processes.

The quantitative evaluation of the Speckle interferometric measuring is carried outaccording to the well-known phase-shift principle [14]. The primary results consist inthe phase differences (Figure 13.11), from which the field distributions of the singledisplacement components can be derived (Figure 13.12). The strain distributions are ob-tained from the displacement field by numerical differentiation.

An optical differentiation can be realized by the shearographic principle. Due tothe relative shift of the interfering paths of rays, only a small number of correlationfringes can be obtained in comparison to the previously mentioned displacement mea-


306

Figure 13.10: Correlation fringes.

Figure 13.11: Phase pictures of x-, y- and z-displacement.

suring. This strain information is overlapped by large geometric influences and slopeinfluences. Therefore, no on-line observation of the strain is possible. Studies haveshown 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 construc-tion testing requires a compact and transportable measuring head, which can be directlyadapted on a testing machine.

The developed and practically tested measuring instrument is based on the applica-tion of modern optoelectronic elements such as laser diodes as a light source, piezo crys-tals for a nanometer exact phase shift of the light and a CCD-camera [15]. For every dis-placement component, a path of rays of illumination with a laser diode, a phase shift de-vice and a shutter is built in the measuring head. The three displacement directions arerecorded nearly simultaneously by rapidly switching between the illumination direc-tions. 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 de-vice as well as the programming of the software to control and evaluate the measuringdata 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, aninitial value for the phase order is needed. This is given in the easiest case manually byon-line observation. In principle, the heterodyne method can be used for the automationof the order determination. It is based on using two light sources of different wave-


307

Figure 13.12: Lines of constant displacement for x-, y- and z-direction.

lengths. For this technique, the measuring head has been expanded to two illuminationsources for each path of rays. However, this procedure requires, in addition to a veryhigh degree of accuracy for the phase determination, also a stronger protection againstdisturbing 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 differ-

entiation the strains of the observed object surface.


308

Figure 13.13: 3-D-ESPI measuring head.

Figure 13.14: 3-D-ESPI on a testing machine.

13.6 Application Examples

The application possibilities of the object-grating method and the electronic Speckle in-terferometry are introduced by three examples. They are related to their use in deforma-tion- 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 wasdone with a notched tensile specimen (Figure 13.15) according to the set-up in Figure 13.7at 650 �C. The task was to determine its plane deformation by using the 2-D object-grating method. To correct possible virtual deformations, the specimen was furnishedwith 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 tipof a fracture mechanic CT-specimen (Figure 13.18). Since in addition to the in-planestrain, also the out-of-plane displacement was searched, the 3-D object-grating methodwas used.


309

Table 13.2: Specifications of the 3-D-ESPI.

Measuring surface: 10 ×7 mm2 to 600 ×450 mm2

Measuring area out-of-plane: 0.4 � � �20 �min-plane: 1 � � �50 �m

Accuracy out-of-plane: 0.04 �min-plane: 0.1 �m

Strain resolution: ca. 10–6

Local resolution: 768 ×580 Pixel or 1024 ×1024 PixelObject distance: 100 mm to 2000 mmMeasuring head dimensions: 250 ×250 ×350 mm3

Displacement measurements qualitative: on-linequantitative: 3-D by phase evaluation

Figure 13.19 shows the calibrated testing set-up with two CCD-cameras directedat the specimen. In Table 13.4, the essential testing data are listed.

From the recorded local vectors describing the different form states, the displace-ment- and strain fields were derived. Figure 13.20 shows the searched out-of-plane dis-placement 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 pressurebutt welding Copper-Aluminium specimen (Figure 13.22), the electronic Speckle inter-ferometry was applied. The ESPI shown in Figure 13.14, which was adapted to the ten-sile machine, was used.


310

Figure 13.15: Tensile specimen; a) incl. reference object; b) geometry.

Table 13.3: Data from the tensile test at high temperature.

Temperature: 650 �CMaterial: Steel: X2CrNi18 9Measuring method: 2-D object-grating method (p=0.2 mm)Dimensions: H=100 mm, W=13 mmTesting field: 19 mm×12.2 mmMaterial behaviour: elastic/inelasticDisplacement measurement: in-plane

Since the measurement of all three displacement directions is possible for smallload intervals, the deformation of the elastic state could be observed on-line and quanti-tatively 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 dis-placement) of an elastic, and in Figure 13.24 of an inelastic deformation are shown.

Finally, the displacement fields were determined by evaluating the phase imagesaccording to the mentioned phase-shift principle, and the 2-D strain distributions were


311

Figure 13.16: Section of the tensile specimen; a) non-deformed; b) deformed.

Figure 13.17: Lines of constant strain a) in x- and b) in y-direction.


312

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: AlMg3

Measuring method: 3-D object-grating method (p=77 �m)Dimensions: W=20 mm, B=2 mmTest field: 6 ×4.5 mm2

Material behaviour: inelasticDisplacement measurement: in-plane, out-of-plane

derived from these. Figure 13.25 shows the isolinear representation of these strainsfrom the inelastic deformation.

13.7 Summary

With the optical field-measuring methods, object values can be determined two- orthree-dimensionally. Therefore, they are used especially when contours, deformationsand 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 well-known for a long time, they were first able to be used when compact lasers for the gen-eration of coherent light and efficient PCs including adapted software for digital imageprocessing of a large quantity of optical measuring data were developed.

13.7 Summary

313

Figure 13.20: Out-of-plane displacement field.


314

Figure 13.21: �x-strain field.

Figure 13.22: Geometry of the Cu-Al specimen.

Optical field-measuring methods are primarily used today in the manufacturing-and quality control as well as in the material- and construction testing. In the manufac-turing- and quality control, they are used among other things for the automatic record-ing of form dimensions and to recognize global or locally limited defects. In the materi-al- and construction testing, these methods are preferably used for determining displace-ment- and strain fields when testing objects with complicated structures with respect totheir dimensioning.

In connection with the development of material laws for the description of the in-elastic behaviour of metallic materials, especially the object-grating method and theelectronic Speckle interferometry have been further developed. Here, the main goal wastheir adaptation for the solution of three essential tasks: Firstly, they should be used forthe observation of inelastic processes in order to get knowledge for the development ofsuch material laws. Secondly, parameters had to be measured for these laws. Finally,experimentally determined displacement- and strain fields were required as a compari-son to the corresponding achieved data obtained by Finite-Element calculations in orderto verify the material laws included in them.

13.7 Summary

315

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.

The essential result of the further development of the object-grating method andthe Speckle interferometry for material- and construction testing consists in realizingcompact measuring instruments, which can be adapted directly on a testing machinethereby making measurements directly in the testing field possible. This was achievedby applying modern optoelectronic elements such as fibre optics, laser diodes andCCD-cameras. In addition, both elastic and inelastic processes can be recorded fromroom temperature up to high temperatures (approx. 1000 �C). The measurement ismade without contact and interaction. The methods yield primarily the field of the 3-Dlocal 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 mea-suring method, which is now available, is based on well-known interferometric paths ofrays, which have been integrated in a compact 3-D system here.

The further developed field-measuring methods have, in the meantime, been usedmany times in various ways in material testing with respect to their reliability.

The results obtained and experiences made have encouraged further tests. Itshould be tested in this way whether these field methods are also suitable for an on-line measurement with the goal of process control. Furthermore, it is planned to modifythe methods so that larger object surfaces can be recorded in order to e. g. carry out anautomated construction or building supervision.


316

Figure 13.25: Lines of constant strain a) in x- and b) in y-direction from the inelastic deformation.

References

[1] R. K. Erf: Speckle Metrology. Academic Press, INC., London, 1978.[2] R. Ritter: Messung von Weg und Dehnung mit Feldmeßmethoden. Materialprufung 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: Moir�everfahren. In: C. Rohrbach (Ed.): Handbuch fur experimentelle Spannungs-

analyse. VDI-Verlag, Dusseldorf, 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. Dummler-Verlag,Bonn, 1986.

[7] V. Cornelius, C. Forno, J. Hilbig, R. Ritter, W. Wilke: Zur Formanalyse mit Hilfe hochtem-peraturbestandiger 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 Hochtem-peratur-Low-Cycle-Fatigue. Zeitschrift fur Metallkunde 81(11) (1990) 783–789.

[9] D. Winter: Optische Verschiebungsmessung nach dem Objektrasterprinzip mit Hilfe einesflachenorientierten Ansatzes. Dissertation TU Braunschweig, 1993.

[10] D. Bergmann, R. Ritter: 3D Deformation Measurement in Small Areas Based on GratingMethod and Photogrammetry. SPIE’s Proceedings Vol. 2782, Besancon, 1996, pp. 212–223.

[11] J. Thesing: Entwicklung eines Versuchsstandes und eine Auswertestrategie zur dreidimen-sionalen Verformungsmessung nach dem Objektrasterprinzip. Studienarbeit am Institut furTechnische Mechanik, Abteilung Experimentelle Mechanik, TU Braunschweig 1995 (un-published).

[12] A. Felske: Speckle-Verfahren. In: C. Rohrbach (Ed.): Handbuch fur experimentelle Span-nungsanalyse. VDI-Verlag, Dusseldorf, 1989, pp. 372–397.

[13] R. Jones, C. Wykes: Holographic and Speckle Interferometry. Cambridge University Press,1983.

[14] D. Bergmann, B.-W. Luhrig, R. Ritter, D. Winter: Evaluation of ESPI-phase-images withregional discontinuities: Area based unwrapping, SPIE’s Proceedings Vol. 2003, Interfero-metry VI, San Diego, 1993, pp. 301–311.

[15] J. Hilbig, K: Galanulis, R. Ritter: Zur 3D-Verformungsmessung mit einem ElektronischenSpeckle Pattern Interferometer (ESPI). VDI-Berichte Nr. 882, 1991, pp. 233–242.

References

317

14 Surface-Deformation Fields from Grating PicturesUsing Image Processing and Photogrammetry

Klaus Andresen*

14.1 Introduction

Grating methods provide a well-known technique for deriving the shape, the displace-ment or the deformation of the surface of an object [1]. A regular periodic grating maybe projected or fixed on the surface of an observed object. If the surface is flat, a sin-gle image is sufficient to derive the physical grating coordinates on the object. For acurved surface, two or more images taken from different locations are needed to calcu-late spatial coordinates by photogrammetric methods. In the images, the coordinates ofsuitable marks will be determined. Depending on the experimental set-up, the physicalcoordinates of a plane surface or the 3-D coordinates of a curved surface will be de-rived from these data [2].

According to the application, a different type of grating is applied, e.g. a line grat-ing, a point grating, a cross grating or a circle grating. Here, cross gratings generatedby two mainly orthogonal bands of lines will be investigated because the related im-age-processing methods proved to be most stable when analysing largely deformedgrating patterns by line-following algorithms. Moreover, the cross point coordinatescould be determined in most cases automatically with subpixel accuracy [3, 4].

Projected gratings provide a simple and cheap means to deliver cross points ofthe surface if only the shape of the object is asked for. However, if the deformation isneeded, only a fixed grating is applicable because the displacement of material pointsfrom an undeformed and a deformed state must be given. The experimental set-ups anddifferent techniques of fixing cross gratings on a surface will be explained in the nextsection.

The accuracy of the grating coordinates in the images of roughly 0.1 pixel limitsthe possible applications. Depending on the resolution of the digitized image and onthe 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 of20 �m is obtainable in an object region of 100 � 100 mm2. The local accuracy of the

318

* Technische Universitat Braunschweig, Rechenanlage des Mechanik-Zentrums,Schleinitzstraße 20, D-38106 Braunschweig, Germany



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 linewidth is about 5–7 pixel, which is an optimal assumption. This means that only inelas-tic deformation of metals in a range larger than 0.01 or 1 percent strain delivers a suit-able accuracy.

Whole-field methods for elastic strains are based on interference or Speckle tech-niques. The related optical patterns need quite different image-processing methods ase.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. Theirbasic patterns are points given by filled circles, crosses given by two intersecting bandsof lines or overlapping circles for very large deformation of sheet metal. Each patternmay be distorted to a certain extent during the deformation process. The coordinatesare principally defined as the centre of the circle or by the intersection of the two armsof 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 rectan-gular matrix of e.g. 512 rows (index x) and columns (index y), respectively. Each ele-ment contains a grey or intensity value (0 � � � 256� 8 bit), which is proportional to thelight intensity of a related small region of the object surface.

The grating coordinates in the images are expressed in pixel. By suitable filteringtechniques, subpixel accuracy is reachable even in noisy images with low contrast. Fora simulated image in Figure 14.1 with a relatively large deformation of the grating, thefrequency distribution of the errors [pixel] is given in Figure 14.2. The results are de-rived with a line-following filter as described in the next section. Obviously, about85% of the deviations from the theoretical coordinates are less than 0.1 pixel. Thelarger 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 crosscoordinates with subpixel accuracy [4]. Such a filter will be constructed according toan idealized intensity distribution of a cross, where the filter constants cij are propor-tional to that distribution. Then, a filtered value �fkl is calculated by a convolution sum:


319

�fkl ��K�2

i��K�2

�L�2

j��L�2

cijfk�i�l�j � �1�

If this filter is applied to a picture, a smooth correlation function is resulting, whichmainly amplifies the cross region and which shows its maximum values in the centreof each cross.

The pixel indices (xm� ym) of the maximum points could be taken as cross coordi-nates, however, with a limited accuracy of one pixel. This can be improved if the maxi-mum ��xmax� �ymax� of a local 2-D polynomial of the form

f ��x� �y� � a0 � a1�x � a2�y� a3�x2 � a4�x�y� a5�y

2 �2�

14 Surface-Deformation Fields from Grating Pictures Using Image Processing

320

Figure 14.1: Simulated cross grating.

Figure 14.2: Frequency distribution of deviation [pixel].

is calculated, which approximates the grey values of the central maximum point and its8 neighbouring points using local coordinates ��x � x � xm� �y � y� ym�. This techniquegenerally provides an accuracy of � 0�1 pixel, and it was applied to different deforma-tion processes showing relatively small changes of the original rectangular cross grat-ing structure.

14.2.2 Line-following filter

For highly deformed cross gratings, the above described technique failed because ofthe big change of the width and the inclination of the crosses and its arms. For theseimages, a line-following filter was developed [6]. It is used to determine both bands ofcross lines separately with subpixel accuracy. The intersection of these two bands thendelivers the wanted cross coordinates with high accuracy. This technique proved to bestable even for strongly deformed and curved grating lines.

For initializing the line-search procedure, first, one point on the line and the relatedline direction must be given. This is performed manually by two click points perpendicu-lar to the line or automatically with a rotational invariant filter [7], which determines thecentre of the line and its direction. But this filter is not stable when passing through crosspoints. Hence, a new elliptic correlation filter of the following form was developed:

G�x� y� � Gx�x�Gy�y�Wy�y� � cos�

2Ax cos

3�2B

y cos�

2By � �3�

A and B in Equation (3) may be regarded as the principal semi-axes of an ellipse andhence as half the filter length and the filter width, respectively (Figure 14.3). This filteris rotated locally into grating-line direction (Figure 14.4). The hat-like filter form Gx inx-direction amplifies all values on the line and the relatively large extension in that di-rection 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 sim-ilar to the central lobe. The negative side lobes provide a zero filter response if appliedto a constant grey level region in the image. The weight function Wy decreases thefunction Gx steady to zero at y � �B, which provides a smooth filter response.


321

Figure 14.3: Filter function of an elliptical filter.

This filter is moved perpendicular through a line in 5 � � � 7 discrete steps. The re-sulting filter response takes on a maximum value in the centre of the line. To providesubpixel accuracy, the filter responses are approximated by a second order polynomialin the maximum point. The maximum value of the polynomial then delivers coordi-nates with subpixel accuracy. A similar method determines the centroid of an area be-tween the filter responses and a suitable threshold to define the centre coordinates of aline. In practice, the filter is moved in column or line direction in the image accordingto which direction is closer to the perpendicular direction because a shift in an arbitraryangle through a grating line needs complex interpolation procedures.

For optimal results, the filter width 2B should be about 2 � � � 2�5 times the grating-line width W and a filter ratio A�B � 2 � � � 2�5 with the larger values for noisy dataguarantees a stable and robust line-following characteristic even through cross pointsand small gaps in the line. The needed line direction for the filter rotation is derivedfrom the foregoing points by extrapolation.

To demonstrate the power of the described technique, a low quality image of thesurface of a metal block, deformed by forging, is evaluated (Figure 14.5). The original-ly rectangular grating pattern of equal line width, etched into the material, becomesstrongly curved in some regions. Also, the line width and the intensity distribution ofthe lines are quite different in horizontal and vertical direction according to the com-pression and extension of the material. Hence, different filters must be used for eachline. Figure 14.6 directly shows the grey values of a small subsection marked in Figure14.5 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. Ob-viously, the centre lines are smooth even when the image is noisy and when the linepattern has a very low contrast.


322

Figure 14.4: The rotatable elliptic line searching filter.


323

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.

14.3 3-D Coordinates by Imaging Functions

When looking at the 3-D displacement of small flat deformation fields – e.g. crack tipsin a volume of 10 � 10 � 2 mm3 –, a simplified numerical method can be used insteadof stereo photogrammetry to derive the spatial grating coordinates [8]. A calibratedgrating on a glass plate is moved exactly parallel and perpendicular to its plane in pre-cise 3 � � � 5 steps �Z in Z-direction. The coordinates �Xijk� Yijk�Zijk� form a dense rectan-gular grid in space, and they are known exactly. Since from each step, an image is re-corded, also the related image coordinates are known. Hence, a pair of 3-D polyno-mial-imaging functions � � f �X�Y �Z�� g�X� Y�Z� for each camera can be calcu-lated, which transforms each grid point �X�Y� Z� into an image point �� , i.e. it ap-proximates the real stereo-imaging function. Suitable approximating functions are poly-nomials with free parameters since that provides a linear-equation system when using aGaussian least-squares fit. Hence one has:

��a� ��

aijkXiYjZk � ��b� �

�bijkX

iYjZk � �4�

where the vectors a � �aijk�� b � �bijk� describe free parameters. Each vector is deter-mined separately by minimization. For a, one has:

��ijk�measured � �ijk�a��2 � min � �5�

A similar equation holds for ��b�. For each camera, this supplies a set of two func-tions, which transform any point within the grid volume into an image point. Vice ver-sa, also a point in space �XP� YP� ZP� can be calculated if its image coordinates��1m� �1m�, ��2m� �2m� are known in at least 2 images. Then, one has:

�1�XP�YP�ZP� � �1m � �2�XP�YP�ZP� � �2m � �6��1�XP�YP�ZP� � �1m � �2�XP�YP�ZP� � �2m � �7�

These are 4 equations for the 3 unknown spatial coordinates �XP�YP�ZP�, which easilyare solved by numerical iteration.

This technique also works if the image plane in the camera is tilted (Scheimpflugcondition), which provides a larger, well-focussed area in space when using stereo cam-eras. Moreover, the method is easy to program and there proved to be no convergenceproblems. It was applied to the propagation of a crack tip [9, 10].


324

14.4 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 adoptedfrom close-range photogrammetry. A measuring device was developed consisting of 2or 3 stereo cameras in a stiff framework. A movable support, which first holds a cali-brating glass plate later on holds the considered objects [11, 12]. Before any measure-ment can take place, the exterior and the intrinsic orientation of the cameras must beknown. 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 coor-dinate system �X� Y�Z�; �X� Y� in the plane and �Z� perpendicular to it. With respect tothis system, the exterior orientation – the projection centre �X0�Y0�Z0� and a rotationmatrix R describing the rotation of the local camera system �x� y� z� into the global sys-tem – must be determined. The constants of the intrinsic orientation are the focallength c, called camera constant, and lens distortion factors A�B�R0, described later on.

The glass plate is moved in 3 parallel steps in �Z -direction, which might be in-clined by small angles �x� �y against the Z-direction; in Figure 14.8, only a plane con-figuration is shown. Given the pitch �X��Y of the cross grating and an arbitrary shiftof the origin �X0�Y0� in that plane, the coordinates of the spatial grid coordinates are:

Xijk � XS � i�X � axZk � �8�

Yijk � YS � j�Y � ayZk � �9�

Zijk � Zk �10�


325

Figure 14.8: Set-up for camera calibration, plane configuration.

with i � 1� � � � �M� j � 1� � � � �N in the plane and k � 1� � � � �L in shift direction, where

ax � tan�x, ay � tan�y and Zk � Z k�

��1 � a2

x � a2y

�� which is the related distance on

the Z-axis due to a parallel shift of Zk. XS�YS��X��Y are given parameters of the cross

grating, while �x� �y, and Zk�k � 1� � � � �L� 1� are unknown quantities, which will bedetermined together with the parameters of the camera orientation in a modified bun-dle-block adjustment. Instead of moving the cross grating, it is also possible to shift thecameras 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 cam-era takes the images of a spatial grid, which approximately coincides with the measur-ing 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 adjust-ment because it is well-known in the relevant publications [2]. The intrinsic and exter-ior orientation of a camera in space is described by its projection centre �X0�Y0�Z0�,the focal length c, the rotation matrix R � �rij� usually given by 3 Euler angles, andthe distance ��0� �0� of the origin in the image plane to the optical axis. Then the trans-formation from the space coordinates �X�Y �Z� to the image coordinates �� is foreach point (i� j� k) within the grid:

� � �0 � c�X � X0�r11 � �Y � Y0�r21 � �Z � Z0�r31

�X � X0�r13 � �Y � Y0�r23 � �Z � Z0�r33� u�� 11�

� � �0 � c�X � X0�r12 � �Y � Y0�r22 � �Z � Z0�r32

�X � X0�r13 � �Y � Y0�r23 � �Z � Z0�r33� �� 12�

u�� describe the lens distortion, which are assumed to be radial symmetric:

u�� A1�R20 � R2� � A2�R4

0 � R4�� 0� � �13�

�� A1�R20 � R2� � A2�R4

0 � R4�� 0� � �14�

where the radius R is given by:

R �� 0�2 � �� 0�2

�� 15�

R0 is a constant of the objective, usually about 70% of the maximum width of the im-age plane, and A1�A2 are the required distortion parameters.

Now p will be defined as the vector of the unknown parameters:


326

p � �X0�Y0�Z0� �0� �0� rij� c�A1�A2�l� ax� ay� Z1� � � � �ZL�1� Xijk�Yijk�Zijk�m� � �16�

where subscript l means that this set of parameters is repeated for each camera. Furtheron, a certain number m of spatial coordinates are not used directly, but they are takento be unknown parameters. This provides a higher accuracy because it couples the pa-rameters of the cameras in a global least-squares fit. ��ijkl� �ijkl� are measured image co-ordinates of the camera l. By a bundle-block adjustment, the unknown parameters in pare determined altogether by minimizing the sum of the squared differences betweenthe measured and the calculated image coordinates. Here, the expression for � is given,a similar expression holds for �:

F�p� ��camera

l

�grid

ijk

��ijkl�p� � �ijkl�2 � min � �17�

To solve Equation (17), it is linearized with respect to an initial parameter vector p0:

F�p0� ��F�p0��p

� p � 0 for every �i� j� k� l� �18�

yielding an overdetermined system of linear equations for an increment � p, which issolved by the least-squares method. Starting from suitable initial values p0, a globaliteration is performed until � p is less than a given threshold. p and its standard devia-tion are the final result. Numerical experience has proved that about half the number ofgrating points should be dealt with as unknown points to get an optimal convergenceand accuracy of the non-linear iteration process.

Programming of the bundle-block adjustment and also its application requires alot of experience, especially choosing suitable initial values becomes a very sensitivetask. 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] todetermine spatial object points by ray intersection, provided the coordinates of the ad-joined 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 know-ing adjoined points if the elements are to be described analytically by a set of free pa-rameters [14, 15]. Practical examples are circles, straight lines and curved lines. Alsocylinders and spheres in space can be treated if a sequence of contour points in theimages are detected.


327

14.5 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 thedisplacement of the vectors dr1

i �i � 1� � � � � 4� in the deformed state and dr0i in the un-

deformed 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 de-formed element will be shifted parallel into the related undeformed centre. Then, thevariation of the four dr-vectors describes a rotation and the desired plastic deformationor strain. The rotation will be separated from the strain in a theory of large deformationas follows. Assuming, the vectors �dr0

1� dr02� will be deformed into �dr1

1� dr12�, then a de-

formation gradient F is calculated from the linear relations:

dr11 � Fdr0

1 � dr12 � Fdr0

2 � �19�

F will be split into the left rotation tensor R and a right deformation tensor U:

F � RU � �20�

From the right Cauchy-Green tensor G:

G � FTF � UTRTRU � UTU � �21�

a deformation tensor [16] is given:

U ��G

�� c01� c1G �22�

with

c1 � 1��trG� 2

��detG

�� c0 � c1detG � �23�

trG � g11 � g22 is the trace of G and detG is the related determinant. The elements ofthe deformation tensor U

U11 U12

U21 U22

� �� 1 � �x �xy

�xy 1 � �y

� ��24�

include the well-known plane-strain components �x� �y� �xy. In a cross point, 4 strain val-ues according to the four meshes surrounding the point are calculated. The average ofthese 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 relativelysmall within the considered area. Then, 4 pairs of vectors as in the plane case are taken


328

to calculate the strain in its centre point. After moving the deformed centre point andthe related vectors into the undeformed state, each of the pairs of spatial vectors, form-ing a triangle, are rotated into an arbitrary reference plane, in which a plain-strain ten-sor 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 islarge. Then, virtual strains arise already from the rotation of an element into a referenceplane.

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 adeformation gradient for the central point using the 8 neighbouring points in the grat-ing.

To derive the deformation function, 4 meshes of a plane grating are consideredwith basic coordinates �x� y� z� as shown in Figure 14.9. The coordinates of the unde-formed grating are �xij� yij� zij � 0� with indices (i� j � �1� 0� 1) or written as a vectorxij � �xij� yij� 0�T. The coordinates of the deformed element are �xij � ��xij��yij��zij�T.


329

Figure 14.9: Undeformed grating element and spatially deformed one.

The total displacement of each point again is given by a rigid body translation, arotation and a plastic deformation of the element. To eliminate the rigid body motion,first, the central vector �x00 will be subtracted:

�xij � �xij � �x00 � �25�

Then, a rotation matrix will be determined, which moves the normal vector n of thesurface element into the z-axis. Approximately, the normal vector is derived from thecross product of the difference vectors:

dx � �x10 � �x�10 � dy � �x01 � �x�0�1 � �26�

n � dx � dy � �27�

The unit vector n0 delivers the 3. row of the rotation matrix R. The first row is takenfrom the unit vector d0

x, which means that the deformed x-direction nearly coincideswith the related undeformed direction after rotating the element. The 2. row is given bythe cross product n0 � d0

x .Now, the rotated coordinates are:

�xij � R�xij �i� j � �1� 0� 1� � �28�

and the related displacement vectors:

qij � �xij � 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 � x � f �x� y� � �y � y� g�x� y� � �z � h�x� y� � �30�

The functions f � g will be approximated by polynomials of second order, e.g.:

f �x� y� � f0 � f1x � f2y� f3x2 � f4xy� f5y

2 �31�

with f0 � 0, since �x00 � 0. For g, a similar function with coefficients gk is taken. Thedeformation function h for �z is less relevant since only the deformation in the tangen-tial plane is considered.

From these functions, the deformation gradients in the central point x00 are de-rived. Hence, one has:

��x��x � 1 � f1 � ��x��y � f2 � �32��y��x � g1 � ��y��y � 1 � g2 � �33�

To determine the coefficients fk�k � 1 � � � 5�, a least-squares method applied to the x-component of displacement vector qxij requires


330

�ij

qxij � f �xij� yij��2�� min � �34�

In the case of rectangular meshes with pitches �x��y in the basic plane, the matrix ofthe normal equations can be calculated analytically. With respect to symmetry, the coef-ficients f1� f2 are totally uncorrelated and one has:

f1 ��ij

xijqxij��ij

x2ij � f2 �

�ij

yijqxij��ij

y2ij � �35�

and similar equations for g1� g2:

g1 ��ij

xijqyij��ij

x2ij � g2 �

�ij

yijqyij��ij

y2ij � �36�

Generally, weight coefficients wij may be introduced according to the significance ofthe displacement values qij, leading e.g. to f1 ��

ijxijqxijwij�

�ijwijx2

ij. If the totally 8

edge points are existing, the following simple expressions are resulting (wij � 1):

f1 � �qx11 � qx10 � qx1�1� � �qx�11 � qx�10 � qx�1�1��6�x� � �37�f2 � �qx11 � qx01 � qx�11� � �qx1�1 � qx0�1 � qx�1�1��6�y� � �38�

Similar equations hold for g1� g2 with qxij replaced by qyij. Taking weight coefficientswij � 0 in the four corner points, the simple central differences, e.g. f1 ��qx10 � qx�10��2�x, are resulting. This proved to yield the least errors for noise-freegrating coordinates as shown in the next section.

Now, the deformation gradient F

F � 1 � f1 f2g1 1 � g2

� ��39�

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 longerrectangular. Then, both elements, undeformed 1 and deformed 2, will be shifted intothe origin. There, they are rotated as described in the foregoing section, meaning thatthe normal vector n1�n2 coincide with the z-axis and that the deformed x-directionsshow into the basis x-direction. Then, a displacement vector is calculated from the dif-ferences of the rotated coordinates:

qij � �x2ij � �x1

ij � �40�and according to Equations (30) to (34), a system of normal equations can be deter-mined. In this case, Equation (34) must be built up and solved numerically because nosymmetry of the meshes is provided.


331

14.6.2 Correcting the influence of curvature

Numerical simulation, as described in the next section, shows a systematic error for thestrain. In the average, it is always too small because only the tangential projection ofthe curved surface is used. A noticeable improvement is derived when the arc length of


332

Figure 14.10: Correcting the displacement vector by arc length.

Figure 14.11: Hat-like deformed metal sheet.

Figure 14.12: Principal strain �I.

deformed surface is considered in the x-z- or y-z-plane, respectively. This can be per-formed approximately by calculating a second-order polynomial of displacement in z-direction through the deformed but translated and rotated points �x�10, �x00, �x10 in Fig-ure 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 dis-placement in x-direction. A similar procedure holds for the y-direction. The errors ofthe 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 defor-mation functions u�r� and w�r� in z-direction were assumed to be radial symmetric:

u�r� � cu sin �2�r�R0� � w�r� � cw cos ��r�R0� � �41�


333

Figure 14.13: Difference of �Ith � �I without compensation of curvature.

Figure 14.14: Difference of �Ith � �I with 70% compensation of curvature.

r ��x2 � y2�

�� arctan �y�x� � �42�

Then, the deformed hat is given by:

�x � x � u�r� cos �� y � y� u�r� sin �� z � w�r� � �43�

The exact principal strains in r-direction are:

�Ith ��1 � du�dr�2 � �dw�dr�2

�� 1 � �IIth � u�r � �44�

Regarding the following figures, the parameters were cu � 0�5, cw � �4�R0 � 5��x � �y � 0�33. In Figure 14.11, the hat-like deformed sheet is given for30 � 30 lines. Figure 14.12 shows the principal strain �I calculated according to Sec-tions 14.5 and 14.6.1. Figure 14.13 demonstrates the related error �Ith � �I if no com-pensation of the curvature is taken into account. Obviously, the difference is alwayspositive with an average value of 0.006.

Figure 14.14 shows the same deviation with 70% curvature compensation due toSection 14.6.2. The average value is reduced to 0.002 and the maximum error from0.011 to 0.007. Hence, in this example, the maximum deviation from the theoreticalvalues is always less than 0.7% of the maximum strain �I � 0�96. A 100% compensa-tion 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 er-ror for the above chosen example. The error increases with the pitch in the interesting


334

Figure 14.15: Relative maximum error of strain �Ith � �I with respect to grating pitch.

interval approximately like a second-order polynomial. For a 10 � 10 grating, the relat-ed maximum error becomes already 6%.

14.7 Conclusion

The grating techniques, as described in this chapter, were applied to a large variety ofdeformation processes, e.g. crack-tip propagation, low-cycle fatigue, high-temperaturecreep, 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 per-formed mainly for research purposes to get insight into deformation processes, to testFinite-Element results or to determine the parameters in new constitutive laws for in-elastic deformation [19, 20].

With some modifications, the photogrammetric equipment and the software alsowere applied to measure the dimension of industrial parts [8] to determine the radius ofcutting tools and to derive the contour of a fossil fish.

A new field of practical applications comes from metal forming. Especially forsheet metal forming of the body works of cars, the grating methods are used for failureanalysis. Often, it must be decided whether the characteristics of the material or theshape of the pressing tool cause the defects on the surface. Also, the influence of oilfilms on the flow of the material and on the friction between tool and sheet metal shallbe investigated to optimize the forming process and the mechanical equipment.

For these future applications, the image-processing hardware and software mustbe 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 computingpower still increases every year, thus allowing more sophisticated algorithms for auto-matic image processing.

References

[1] P. J. Sevenhuijsen, J.S. Sirkis, F. Bremand: Current trends in obtaining data from grids. Ex-per. Techn. 27 (1993) 22–26.

[2] K. Kraus: Photogrammetrie, Vol. 2. Dummler, Bonn, 1984.[3] J.S. Sirkis, T. J. Lim: Displacement and strain measurement with automated grid methods.

Exp. Mech. 31 (1991) 382–388.

References

335

[4] K. Andresen, B. Hubner: Calculation of Strain from an Object Grating on a Reseau Filmby a Correlation Method. Exp. Mech. 32 (1992) 96–101.

[5] K. Andresen, Q. Yu: Robust phase unwrapping by spin filtering combined with a phase di-rection 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 Linkoping University, Dept. of Electrical Engineering, Linkopin, Sweden, 1983,p. 581.

[8] K. Andresen: 3D-Vermessungen im Nahbereich mit Abbildungsfunktionen. Mustererkennung92, 14. DAGM Symposion, Dresden, 1992, pp. 304–309.

[9] K. Andresen, B. Kamp, R. Ritter: 3D-Contour of Crack Tips Using a Grating Method. Sec-ond 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 bya grating method applied to crack tips. Opt. Eng. 31 (1992) 1499–1504.

[11] K. Andresen, Z. Lei, K. Hentrich: Close range dimensional measurement using gratingtechniques and natural edges. SPIE 2248 (1994) 460–467.

[12] K. Andresen, K. Hentrich, B. Hubner: Camera Orientation and 3D-Deformation Measure-ment 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 fur Photo-grammetrie und Fernerkundung 59 (1991) 212–220.

[15] F. Neugebauer: Calculation of curved lines in space from non-homologues edgepoints. Opti-cal 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. Zeit-schrift der Techn. Univ. Dresden 18 (1969) 531–538.

[18] L. Eberlein, P. Feldmann, R.V. Thi: Visioplastische Deformations- und Spannungsanalysebeim Fliesspressen. Umformtechnik 26 (1992) 113–118.

[19] K. Andresen, R. Ritter, E. Steck: Theoretical and Experimental Investigations of Fractureby FEM and Grating Methods. Defect Assessment in Components – Fundamentals and Ap-plications. Mechanical Engineering Publications, London, 1991, pp. 345–361.

[20] K. Andresen, S. Dannemeyer, H. Friebe, R. Mahnken, R. Ritter, E. Stein: Parameteridentifi-kation fur ein plastisches Stoffgesetz mit FE-Methoden und Rasterverfahren. Bauing. 71(1996) 21–31.


336

15 Experimental and Numerical Analysisof the Inelastic Postbuckling Behaviourof 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 andloading of the aerospace structure shown in Figure 15.1. Although the example in Fig-ure 15.1 depicts a possible structure of a hypersonic vehicle, the construction is typicalfor supersonic and common subsonic transport aircrafts. To examine the basic phenom-ena of the load-carrying behaviour in the postbuckling range, the analysis of such struc-tures can be reduced to a simplified mechanical model of an initially flat, shear-loadedpanel. For practical reasons, our investigations were limited to aluminium (Al2024-T3)panels at ambient temperature and 200 �C. Within the high postbuckling regime or athigh load levels and elevated temperatures in supersonic vehicles, moderate inelasticstrains occur and the behaviour of the considered structure becomes geometric andphysically non-linear.

Due to the high complexity of the problem, analytical investigations must be ac-companied by tests in order to validate the numerical model. It is based on the Finite-Element method and therefore can be easily applied to different geometries and bound-ary conditions. The main problem of the numerical model is the choice of suitablematerial models since no universal material model for the description of the inelasticbehaviour of arbitrary metallic materials exists. To simplify the adaption of the pre-sented numerical model to different materials, the used solution algorithm is designedto allow a very easy implementation of different material models. In case of the consid-ered aluminium alloy, the performance of several material models is examined for therate-independent plasticity at ambient temperature and visco-plastic behaviour at ele-vated temperature. The identification of their parameters from suited material test re-sults is demonstrated.

All shear tests, including quasistatic monotonic, cyclic and creep- and relaxationtests at elevated temperatures, are conducted with a specially designed test set-up

337

* Technische Universitat Braunschweig, Institut fur Flugzeugbau und Leichtbau,Hermann Blenk Straße 35, D-38108 Braunschweig, Germany



(“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 exactlyat the corners of the square specimen. The pins are parted so that the tested area of thespecimen 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 me-chanical model shown in Figure 15.3 are not influenced by uncertainties in the assump-tion of the geometry or the boundary conditions.

The test set-up is equipped with nine infrared radiators in front of the shear paneland eight heating elements, which are placed directly on the edges of the shear frame.In combination with five separate temperature controllers, a nearly constant temperaturedistribution can be achieved in the tested area of the panel up to 200 �C. Until now, 78monotonic and cyclic tests at ambient and elevated temperature and different panelthicknesses have been performed on this test set-up.

15 Experimental and Numerical Analysis of the Inelastic Postbuckling Behaviour

338

Figure 15.1: Typical structure.

Figure 15.2: Test set-up PApS.

15.2 Numerical Model

15.2.1 Finite-Element method

Two partly different formulations of the fundamental equations and affiliated solutionmethods are used. The behaviour of the considered material at room temperature canbe 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 geome-trically non-linear static equations of the plate theory, and the problem is solved by incre-mental-iterative methods. At the other hand, the visco-plastic problem introduces realtime-dependency with higher demands regarding the time-integration accuracy and stabil-ity. Therefore, a more closed formulation of the fundamental equations of the continuumtheory 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 atotal Lagrange formulation with an additive decomposition of the strain tensor, large de-formations but only small strains are admissible. The mixed variational principle, which isused to derive the finite elements, bases on the Kirchhoff-Love plate theory. This theoryyields reasonably good results since the considered panels are sufficiently slender. All ma-terial models are implemented in the numerical model by means of the normality rule ofthe classical theory of plasticity, applying a v. Mises type of inelastic potential. A mixedprinciple is chosen since such a formulation produces displacements and stresses with thesame degree of approximation as primary unknowns, and therefore provides some advan-tages concerning the computational effort in the treatment of the non-linearities. All pri-mary unknowns of the described variational formulations are approximated with bilinearpolynoms, yielding a four-noded plate element.


339

Figure 15.3: Mechanical model.

15.2.1.1 Ambient temperature – rate-independent problem

The used variational functional is derived from the principle of virtual displacementswith the strain-displacement relation as a restriction and reads for a certain state t:

�At � �

�F

�t�� 12�tC�1�

� �dF �

�F

��tp dF � 0 � �1�

This principle is transformed into an incremental form, yielding the linear stiffness ma-trix, the tangent and the secant matrix. The solution for each increment is achieved byan arc length-controlled modified Newton-Raphson iteration. In this form, the materiallaw in the incremental form �� D�� is comprised in the tangent matrix, and thenon-linear constitutive equations of the rate-independent material models can be easilyincluded. The in-plane, coupling and bending stiffnesses of the plate material are deter-mined by integration of the actual elastic-plastic tangent moduli D given by the usedmaterial model over the plate thickness. This integration assumes discrete layers withconstant properties in thickness direction, and the description of the material behaviouris reduced to the plain stress constitutive equation in each layer. The elastic-plastic tan-gent modulus is updated only once for each load increment using the results of the pre-vious load step, yielding an Euler-Cauchy type of integration of the non-linear constitu-tive equations. This method considerably reduces the numerical effort. Furthermore, nodifferentiation of the constitutive equations is needed as it would be the case in implicitintegration methods, and therefore, the material model can be changed very easily. Sta-bility problems in the integration of the used material models were never observed, andit can be shown that the error in the solution for the used constitutive equations re-mains sufficiently small since the magnitude of the load increments, which is restrictedby 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 ef-fects. However, due to the quasistatic basis of the method, a so-called instable equilib-rium path connects the starting- and the end point of the snap-through. This path repre-sents a fair approximation only for moderate snap-throughs. Simulations of severesnap-throughs, which occur in the range of unloading during cyclic shear tests aftervery high load amplitudes, lead – in connection with the development of plastic strainsduring the snap-through – to obviously wrong solutions and numerical problems. Toimprove the capabilities of the used method, the described Finite-Element formulationwas extended to calculate dynamic effects by solving the complete equation of move-ment. For simplicity, the damping matrix is formed by a linear combination of the usedconsistent mass matrix and the system-stiffness matrix. The magnitude of damping isfitted to experimental results. The accelerations and the velocities are derived from thedisplacements using the Newmark scheme. To reduce the numerical effort, the dynamicmethod is only used if the quasistatic method detects a limit point in the loading pathand a snap-through is starting. In this point of loading, the determinant of the systemmatrix changes its sign. When after the snap-through, the velocities of the structure aresmall enough, the quasistatic method is used again.


340

15.2.1.2 Elevated temperature – visco-plastic problem

The temporal derivative of the basic variational principle (Equation (1)) is used to de-rive 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 ofLagrange multipliers. In case of the Chaboche model, those equations are: the consis-tency condition, the overstress function and the evolution equations for isotropic andkinematic hardening. The corresponding Lagrange multipliers can be determined by therequirement that each term of the variational principle has to form an energy. Like inthe rate-independent problem, a layered model is adopted. However, the plate stiffnessis formulated directly in terms of the primary unknowns. In the sense of a “rateapproach” [3], the primary unknowns are the velocities, the temporal derivatives of thestresses, respectively the membrane forces, and the bending moments plus the temporalderivative of the visco-plastic potential and the effective strain rate of each layer. Thecomplete visco-plastic problem is solved by a predictor-corrector method similar to amidpoint-type time-integration algorithm. One predictor and one corrector step is usedfor solving the equations in one time-increment. A comprehensive discussion of thismethod is given in [4]. This time-integration scheme was chosen in order to avoid theanalytical or numerical determination of the tangent-stiffness matrix needed in theframework of implicit time-integration schemes to establish the Newton-Raphson itera-tion. Therefore, this method reduces the difficulties of the implementation of new con-stitutive equations. An automatic time-step control is established by limiting the magni-tude of the increment of equivalent total strain within each time-step. The critical valuefor 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 testsat room temperature and elevated temperature are given in Figure 15.4. There is a dis-tinct plastic orthotropy at room temperature, which must be taken into consideration inthe material model. This orthotropy vanishes at higher temperatures. The results of thetests at 200 �C with different loading rates give an impression of the changed mechani-cal 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 ambi-ent temperature can be described by two or more surface rate-independent materialmodels based on the classical theory of plasticity. In the conducted investigations, theperformance of twelve different material models, respectively combinations of their ba-sic characteristics, were examined.


341

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 influ-ence of the translational rule on the performance material models shall be discussed inmore detail. All of the described models use v. Mises-type yield surfaces. Deformableyield surfaces shall not be discussed here in more detail.

The Mroz model for the material used is sketched in Figure 15.5. Four surfacesare located in the stress space. The inner one surrounds the elastic region and is of theHill-type. This type of yield surface is based on the v. Mises surface, but can be de-formed by adjusting additional shape parameters. All surfaces can move in the stressspace 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 incyclic loading of metallic materials. Furthermore, the description of the plastic aniso-tropy is made possible by adjusting the starting values of the backstress tensor. There-fore, kinematic hardening is necessary even for the simulation of tests with monotonicloading. Plastic loading takes place when the stress point is located on the yield surfaceand moves in an outward direction. Stress states beyond the yield surface are not ad-missible. This is controlled in all types of material models of this category by the so-called consistency condition. Therefore, the movement of the yield surface during plas-tic loading with kinematic hardening is restricted by this condition. The direction of themovement of the yield surface has to be established separately. For the Mroz model,


342

Figure 15.4: Tensile tests at room temperature and at 200 �C.

the translation of the yield surfaces is chosen in a way that subsequent yield surfacesget into contact in points with equal directed normals. In other words, the surfacesapproach each other tangentially.

An isotropic hardening during plastic loading, which means an expansion of thesurfaces, is possible. This type of hardening describes the stabilization of the materialshysteresis and is controlled by means of the accumulated effective plastic strain. Itscontribution to the entire hardening is small since the considered alloy shows a rapidstabilization under cyclic loading. Each surface of the Mroz model is connected with aconstant plastic tangent modulus. This causes a piecewise linear approximation of theinelastic stress-strain relation, which is a central advantage of this model since the ini-tial position of the yield surfaces, their diameter, the shape parameters of the yield sur-faces and the plastic tangent moduli can easily be determined as the graphic in Figure15.5 demonstrates. Unfortunately, the storage capacity needed for this model in numeri-cal 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 onlyone yield surface and a so-called memory surface in the stress space. The yield surfaceshows kinematic and isotropic hardening, whereas the memory surface can only expandin the sense of isotropic hardening. The translational rule for the kinematic hardening ischosen in an approximation of the test results of Phillips [7]. These results indicate thatthe translation on the yield surface is directed along the actual stress increment. Toavoid an intersection of the yield- and the memory surface, Tseng and Lee [8] formu-lated their translational rule as a combination of the “Phillips direction” and a directionof movement, which is determined by the distance measure between the actual stresspoint on the yield surface and a point on the memory surface with the same outer nor-


343

Figure 15.5: Mroz model.

mal. Unfortunately, the Mroz direction predominates the kinematic hardening in someconstellations of the position of the yield surface and loading direction. To improve thisbehaviour, an own modified formulation for a combined translational rule was devel-oped, which enforces the movement along the “Phillips direction”. In both models, thetangent modulus is determined as a function of a distance measure between the actualstress point and a point on the memory surface. Considering non-proportional harden-ing effects, the distance measure used in the equation of Dafalias for the tangent modu-lus is directed along the stress increment. If the yield surface is in contact with thememory surface and the distance measure becomes zero, the tangent modulus is deter-mined by a simple Ramberg-Osgood power law, which is fitted to the properties of thematerial under monotonic loading. The usage of two different equations for the tangentmodulus – for repeated loading and for monotonic loading – is affirmed by tests con-ducted by Phillips [7].

The fourth model in this comparison is formed by a two-surface model with apure Mroz-type translational rule and the Dafalias equation for the tangent modulus.

All parameter identifications for the considered models are performed by meansof stochastic, respectively evolutional optimizing methods, using a least-square formula-tion of the object function. To use the material models for the calculation of the behav-iour of the shear-loaded panels, their parameters are identified simultaneously by the re-sults of four uniaxial tension-compression tests with specimen made of the consideredaluminium alloy in the typical pretreatment state. The results of the tension-compres-sion tests are fairly good approximated by all four described material models. An ex-ample is given for the Tseng-Lee model in Figure 15.4, where only the first tensileloadings are depicted.

Since wide areas of cyclic shear-loaded and buckled panels exhibit non-propor-tional paths for one component of normal strain and the shear strain, special emphasisis laid on the capabilities of the material models in reproducing non-proportional hard-ening effects. Results of strain-controlled tension-torsion test (provided by G. Lange etal. [9], Institute of Material Science, Techn. Univ. Braunschweig) are used to examinethe performance of the material models considering the effect of non-proportional hard-ening. The results of the simulation of a typical non-proportional strain path are shownin Figure 15.6. The first loading of the tubular specimen leads to pure shear. Then, thespecimen is unloaded and a combined tension-torsion loading starts. After total unload-ing, this load cycle starts again. Obviously, the models using a pure Mroz-type transla-tional rule show a poorer correlation with the test data, especially in the developmentof the tensile stress. In contrast, the models using the combined translational rules areable 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 be-haviour of shear-loaded panels at ambient temperature, are calculated with the de-scribed Tseng-Lee material model.

15.2.2.2 Elevated temperature – visco-plastic problem

The visco-plastic problem is treated with unified, respectively overstress material mod-els. The models of Steck [10] (in an isothermal formulation, see Equations (2) and (3))


344

and Chaboche [11] with several modifications are in examination. Several creep, stress-and strain-controlled uniaxial tests were performed (supported by K.-T. Rie et al. [12],Institute of Surface Engineering and Plasmatechnology, Techn. Univ. Braunschweig) toprovide results for the process of parameter identification. This is necessary because ofthe strongly underdetermined character of such a problem, and since it is known that aparameter identification with only one distinct type of test result usually cannot repre-sent the properties of the material sufficiently.

• Steck model:

��in � A1e�V2�

iso �sinh �V1�eff��Nsign ��eff� with ��iso � h1e

�V3�iso ��in� � A2e

V4�iso� �2�

��kin � h2e�V1�

iso�V5�kin sign ��eff��in � A3 sinh �V6 �

kin� and �eff � �� kin � �3�

• Chaboche model:

��in � fK

� �N

� for f � 0 with f�� D�T � X� � R� � X � C23a�in � X ��in

� �

and R� � � �Q� R��in � �4�

Figure 15.7 depicts the results for the simulation of three creep tests with the Steck modeland the basic Chaboche model (Equation (4)). The parameters of both models are simul-


345

Figure 15.6: Results of tension-torsion tests compared to different material models.

taneously identified from the three creep tests. The creep rates of the uniaxial tests are ofthe same magnitude as the typical rates occurring in certain spots of the correspondingshear-panel tests. However, the results of Finite-Element calculations performed withthese sets of material parameters are not in good correlation with the shear tests. It is as-sumed 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 load-ing rates, and a special transient, stress-controlled test are achieved.

The simultaneous identification of the parameters of the material models fromthose tests is performed by optimizing methods using a combination of gradient andstochastic algorithms. Unfortunately, parameter identifications with both models are notsuccessful. Numerical experiments show that the main problem is the reproduction ofthe tension test at high loading rates (the result of a test at 1 MPa/s is depicted in Fig-ure 15.4). To avoid this problem, the overstress function of the Chaboche model ismodified by adding a term accounting for additional, rate-independent inelastic strains.This “overlaying method” is among others described in [13] and [14]. Within the engi-neering approach, the interaction between both parts of inelastic strains is neglected.Good results are achieved with a very simple approximation of those rate-independentstrains by a type of Ramberg-Osgood power law for isotropic hardening. It is the sameapproximation as it is assumed for the monotonic hardening regime within the rate-in-dependent Tseng-Lee model described above. With this additional term, two new pa-rameters are introduced into the material model. A very good starting value (regarding


346

Figure 15.7: Simulation of creep tests with the Steck and the basic Chaboche material model.

further parameter identification of the complete material model) for those parameterscan be found manually within a few iterations if it is assumed that the major part ofthe inelastic strain, occurring in a fast tension test (see Figure 15.4), is described onlyby this Ramberg-Osgood term. The approach without kinematic hardening is possiblehere since no inelastic orthotropy is present. As a matter of fact, no cyclic effects con-cerning this part of inelastic strains can be simulated. The results of a simultaneousidentification of the parameters of this model are given in Figure 15.4 for the tensiontests, 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 straindistribution was provided by R. Ritter and H. Friebe [15], Institute of MeasurementTechniques and Experimental Mechanics, Techn. Univ. Braunschweig) were conductedto characterize the multiaxial behaviour of the considered material and to examine theaccuracy of the material models in the multiaxial case. Figure 15.10 shows the mea-sured 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 speci-men was loaded to a nominal value of tensile stress of 180 MPa in the smallest crosssection. 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 givenin this paper concerning the behaviour of shear-loaded panels at elevated temperatureare calculated with the described modified Chaboche material model.


347

Figure 15.8: Simulation of creep tests with the modified Chaboche model (combined identifica-tion).


348

Figure 15.9: Simulation of transient tests with the modified Chaboche model (combined identifi-cation).

Figure 15.10: Test and numerical results for a creep test with an inhomogeneous specimen.

15.3 Experimental and Numerical Results

15.3.1 Test procedure

All tests conducted on the test set-up PApS at ambient temperatures are incremental-step tests. The temporal course of loading of the specimen at elevated temperature iscontrolled by a computer. In case of tests at elevated temperatures, the specimen ismounted loosely in the shear frame at ambient temperature first. Then, the shear frameand the panel are heated. After that, the 100 screws, which clamp the specimen be-tween the halves of the shear frame, are tightened. This procedure avoids a preloadingof the panel and the occurrence of significant thermal buckles due to the different ther-mal strains of the aluminium panel and the steel shear frame. Nevertheless, the imper-fections of the panel at the beginning of the mechanical loading are slightly higher thanin tests at room temperature. The angle of shear (see Figure 15.3) and the central de-flection of the buckled panel are determined by inductive displacement transducers. Ad-ditionally, 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 engi-neering photogrammetry.

15.3.2 Computational analysis

The shear panels are discretized with regular meshes as it is shown in Figure 15.11. Si-mulations of shear-angle-controlled tests are conducted with prescribed deformations ofthe edges of the panels. The resulting load is obtained by integrating the stress resul-tants along the edges. In case of load-controlled tests like creep tests, the rigid clamp-ing of the panels is simulated by introducing constraints for the nodal deformations onthe edges into the Finite-Element equation system. The load is then applied as a singleforce on one corner of the panel. An examination of the convergency of the spatial dis-cretization of the panels showed that regular meshes with 20 ×20 elements are suffi-cient since a further refinement gives no significant improvement of the results for thecentral deflection (Figure 15.11) or the effective shear load. Since the computational ef-fort – especially for the solution of the visco-plastic problem – is very high even16×16 element meshes are used. Meshes with refinements near the clamped bound-aries of the panel improve the solution only when the whole mesh is very coarse. Ithas been shown by comparative analyses that the idealization of the panel with tenlayers in thickness direction is sufficient for the rate-independent problem. Typicalcreep analyses are conducted with only seven layers without a significant loss of accu-racy in the global and local results since the distribution of the inelastic strains alongthe cross section of the panels is more smooth than in the spontaneous plasticity prob-lem. The determination of the critical time-step sizes related to accuracy and stabilityshall not be discussed here in detail. Numerical experiments show that the creep behav-iour of the shear panels is reproduced within acceptable accuracy for the current param-


349

eters of the material model if a time-step of about 10 s is used at the very beginning ofthe creep process. This comparatively small time-step can be increased very rapidly.The critical time-step for stability can be determined also by numerical experimentsand 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 ofpanels with 1.4 mm thickness at ambient temperature. The angle of shear is plotted ver-sus the central deflection. For an undamaged plate, the first symmetric buckling modealways corresponds to the lowest eigenvalue. For this reason, the plate will buckle sym-metrically. Within the pre- and the lower postbuckling range, the behaviour of the pan-els is strongly influenced by initial geometric imperfections. The influence of the geo-metric imperfections vanishes at least when the angle of shear reaches values of about0.12 �. As the first plastic deformation occurs at an angle of shear of about 0.2 �, thegeometric imperfections do not affect the plastic deformation. Numerous numerical ana-lyses show that the angle of shear at first yielding is approximately a constant for panelthicknesses between 1.2 mm and 3.0 mm.

The first yielding takes place at a spot on the edges, where the main buckle isconstrained by the clamping. Further load increase leads to a propagation of plastic re-gions at this spot and along the tension diagonal on the concave sides of the buckle.


350

Figure 15.11: First loading of shear panels with a thickness of 1.4 mm.

The load-angle-of-shear diagram shows no direct influence of the first plastic deforma-tions on the overall stiffness of the panels until – depending on the panel thickness –the angle of shear reaches values of 0.5 � or more. Figure 15.12 a depicts theoretical re-sults for the deformation state and current distribution of the tangent modulus in threeplanes of the plate for a 1.6 mm panel at different loading states. The tangent modulusis a measure for the local stiffness and the development of plastic strains since it repre-sents the slope of the uniaxial reference-stress-strain curve of the material. Therefore, inthe regime of elastic straining, its value is equal to the elastic modulus and decreaseswith increasing plastic loading. The distribution of the tangent modulus at an angle ofshear of 0.35 � (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 nearlyelastic states. Higher loads lead to a distribution of the tangent modulus, which isshown on the right-hand side of Figure 15.12 a. In this case – at an angle of shear of0.6 � –, the plastic regions cover the whole plate, and in the tension field, the plastic ortangent modulus decreases considerably due to large plastic deformations. From a com-parison of the deformation states follows that the magnitude of the buckling deflectionsare not very much increased from the lower to the higher load as the load-carryingmechanism 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 cyclicallyloaded panel since remaining deformations from prior plastic loading act like geometri-


351

Figure 15.12 a: Deformation states and distributions of the plastic tangent modulus at maximumload.

cal imperfections each time when zero load is passed. The computed distribution of thedeflection and the corresponding distribution of the effective plastic strain at zero loadare shown in Figure 15.12 b. As a matter of fact, the larger remaining deformationsafter higher loads lead to more intense snap-throughs from the current buckling form tothe 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 ofthe angle of shear, are illustrated. Due to inevitable small disturbances in the tests, thedirection 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 complicatedsince it can run through different even unsymmetric buckling forms as intermediatestates. In cases when the central deflection has the same sign in both load extrema, theload reversal can lead to a “double-snap-through” with an intermediate state with oppo-site central deflection. For the theoretical computations, a change of the deformationpath after reaching the bifurcation point at load reversal – the so-called branch switch-ing – is obtained by using small geometric imperfections or disturbances correspondingto the desired buckling mode. In cases of a more intense and complicated snap-throughbehaviour, the described dynamic method is used. In this case, the branch switching ismanaged by applying a suitable distribution of accelerations, which disturbs the systemand 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 tothe plate in a moment corresponding to the depicted intermediate deformation state areshown.


352

Figure 15.12 b: Deformation states and distributions of the effective plastic strain before snap-through after different maximum loads.


353

Figure 15.13: Angle of shear vs. central deflection, results for 1.4 mm panels at different load am-plitudes.

Figure 15.14: Dynamic snap-through, deformations and velocity distribution.

The snap-through in Figure 15.14 was computed in the first cycle of a cyclicshear test (panel thickness 1.6 mm) at the very high amplitude of the angle of shear of0.7 �. The results for the central deflection of this test are given in Figure 15.15. Addi-tionally to the numerical results achieved with the Tseng-Lee model, the starting- andending points of the first and second snap-through computed with the Mroz model areshown in this figure. As can be easily seen, the accuracy of the used material modelhas a large influence on the reproduction of the snap-through behaviour since it de-pends strongly on the remaining plastic deformations at the bifurcation points. In mostcases, the repeated buckling behaviour remains the same after the second load cycle. Inthis test, the cyclic buckling behaviour did not change. Only in one of 22 cyclic tests, arandom behaviour in changing the sign of the central deflection was observed. Further-more, Figure 15.15 shows that the global cyclic load-deformation course is obviouslystabilized after the second cycle, and the results for the 10th and 50th cycle are nearlyidentical. Since the correlations between numerical results and test results are fairlygood, the numerical model is applied to different aspect ratios (a/b, see Figure 15.3). InFigure 15.16, the development of the central deflection and the buckling pattern of twopanels with different aspect ratios are shown. Typical results, which can be used in thedesign of shear panels for defining the distance of stringers and frames, are shown inFigure 15.17.


354

Figure 15.15: Angle of shear vs. central deflection, results for 1.6 mm panels at very high loadamplitude (left) and at higher numbers of cycles (right).


355

Figure 15.16: Buckling modes and central deflection, cyclic loading, aspect ratios a/b= 1.5 and2.0.

Figure 15.17: Load at angle of shear of 0.3 � and 0.5 � vs. thickness, different aspect ratios.

15.3.2.3 Time-dependent behaviour

The general buckling behaviour and the buckling mode of panels loaded at 200 �C arevery similar to those at room temperature. Results for the monotonic first loading ofpanels at different temperatures are shown in Figure 15.18. The loading rate for thetests at elevated temperature is 0.17 kN/s. It is known from the parameter identificationof the material models, that the tangent modulus at the beginning of loading at 200 �Cis only 10% smaller than at room temperature. Therefore, the “global stiffness”, whichmeans the ratio between the global angle of shear and the load, is nearly the same forall examined temperatures in the prebuckling regime. When the panel starts to buckle,the occurring bending stresses together with the shear stress lead to early inelasticstrains, which yield a stronger decrease of the global stiffness than at room tempera-ture. Further analyses show that in the range of panel thicknesses between 1.2 and1.8 mm, an increase of the panel thickness of approximately 0.2 mm covers the loss ofglobal stiffness in the postbuckling regime due to the increase of the temperature from20 �C to 200 �C. The development of the central deflection during monotonic loadingof the shear panels at 200 �C is almost equal to the situation at room temperature if thepostbuckling regime is concerned. The theoretical buckling loads at 200 �C tend to beslightly higher, but due to the undetermined imperfections, a proper measurement ofthis effect is impossible. The influence of different loading rates on the monotonic be-


356

Figure 15.18: Load vs. angle of shear for 1.6 mm panels at different temperatures.

haviour of the panels is up to now examined between 0.17 kN/s and 0.01 kN/s. Allloading rates lead to no significant changes in the monotonic behaviour regarding buck-ling 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 globalstationary creep regime are in good correlation with the measurements. The creep ratesin the primary phase are underestimated by the numerical model, especially at lowerload levels since there is no smooth transition between loading and creep phase. This isnot only a problem of the description of the primary and secondary creep behaviour ofthe material since there is a second more geometric non-linear effect. In this first creepphase, a more rapid relaxation of the bending stresses due to the buckling must takeplace. In the case of spontaneous plasticity, the increase of the load reduces the shareof bending within the whole load-carrying mechanism and the tension componentalong the diagonal increases. During a creep process, this load-carrying state is reacheddue to the permanent generation of inelastic strains. The used numerical model givesbetter results for the first creep phase if this tension load-carrying state is alreadyreached during the loading phase by applying higher creep loads (Figure 15.19). Thedevelopment of the central deflection indicates only small changes in the deformationof the panel during creep. The creep test at a load of 65 kN shown in Figure 15.19yields to an increase of the central deflection of 0.7 mm within the first 360 min. Testsat lower creep rates yield even lower increases of the central deflection.


357

Figure 15.19: Angle of shear vs. time for different creep tests, thickness 1.6 mm.

Figure 15.20 illustrates the influence of creep load and panel thickness on thecreep rates. In this figure, the creep-shear angle is defined as the difference between theshear angle at the start of the creep process and the value after 360 min giving an inte-gral measure for the occurring creep rates.

Finally, Figure 15.21 shows the computed development of the effective inelasticstrain for a creep test at 35 kN. The distributions at the start of the creep process on theleft-hand side are scaled by factor 104, and those at the end of the creep process on theright-hand side of Figure 15.21 by factor 103. In general, the distributions are againsimilar to those of the rate-independent problem. During the creep process, the highestincreases of the inelastic strain can be found at the corners on the lower side and alongthe 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 modelgive a good insight of the behaviour of shear-buckled aluminium panels as far as the be-haviour at monotonic, cyclic and creep loading is concerned. In case of the rate-indepen-dent problem, the Tseng-Lee material model is best suited to simulate the inelastic behav-iour of the material under consideration since this model describes the effect of non-pro-


358

Figure 15.20: Increase of the angle of shear after 360 min.

portional loading very accurately. The visco-plastic problem is treated with the Chabochematerial model. To describe the material behaviour at higher load levels, an additional rate-independent term is added to the Chaboche model. Very high amplitudes of the externalshear loads in cyclic tests require a Finite-Element method with an algorithm accounting fordynamic effects to describe the complex snap-through behaviour. The simulation of thebehaviour of panels with different geometries was performed successfully. The accuracyof the developed method is proved by the fact that very good results are achieved for dur-ability analyses using the output of the numerical simulations as input.

List of Symbols

C matrix of in-plane, coupling and bending stiffnessD yield tensorF area of the plate midsurfacep 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

List of Symbols

359

Figure 15.21: Effective inelastic strain distributions for a creep test at 35 kN; left-hand-side: factor104, t=5 min; right-hand side: factor 103, t=360 min.

a, Ai, Vi, N, parameters of the material modelsK, Q, C, ��in uniaxial inelastic strain� uniaxial stress[� � �]t state t

References

[1] K. Wolf: Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilscha-len aus kohlenstoffaserverstarktem Kunststoff. Inst. f. Flugzeugbau u. Leichtbau, TechnischeUniversitat Braunschweig, 1989.

[2] P. Horst: Zum Beulverhalten dunner, bis in den plastischen Bereich zyklisch durch Schubbelasteter Aluminiumplatten. ZLR-Forschungsbericht 91-01, ISBN 3-9802073-5-8, Inst. f.Flugzeugbau u. Leichtbau, Technische Universitat Braunschweig, 1991.

[3] J.L. Chaboche: A Review of Computational Methods for Cyclic Plasticity and Viscoplastici-ty. Proc. of Int. Conference: Computational Plasticity – Models, Software and Applications,Barcelona, 1987, pp. 379–411.

[4] J. Knippers: Eine gemischt-hybride FE Methode fur viskoplastische Flachentragwerke unterdynamischen Einwirkungen. Berichte aus dem Konstruktiven Ingenieurbau, Heft 18, ISBN3-79831548-5, Technische Universitat Berlin, 1993.

[5] H. Kossira, P. Horst: Cyclic Shear Loading of Aluminium Panels with Regard to Bucklingand Plasticity. Thin-Walled Structures 11 (1991) 65–84.

[6] P. Horst, H. Kossira, G. Arnst: On the Performance of Different Elasto-Plastic MaterialModels Applied to Cyclic Shear-Buckling. Proc. of the Int. ECCS-Colloquim: On the Buck-ling 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. Plas-ticity 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.


[10] H. Schlums, E. Steck: Description of Cyclic Deformation Processes with a Stochastic Mod-el 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 Low-Cycle Fatigue: Description of Deformation Behaviour and Creep-Fatigue Interaction. Thisbook (Chapter 3).

[13] E.-R. Tirpitz, M. Schwesig: A Unified Model Approach Combining Rate-Dependent andRate-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 fur Metalle –Theorie, Numerik und Anwendung. Report 92-70, Inst. of Structural Mechanics, Techn.Univ. Braunschweig, 1992.



360

16 Consideration of Inhomogeneities in the Applicationof Deformation Models, Describing the InelasticBehaviour of Welded Joints

Helmut Wohlfahrt* and Dirk Brinkmann**

16.1 Introduction

The local loads and deformations in welded joints have rarely been investigated underthe aspect that the mechanical behaviour is influenced by different kinds of microstruc-ture [1]. These different kinds of microstructure lead to multiaxial states of stresses andstrains, and some investigations [2–4] have shown that for the determination of the to-tal state of deformation of a welded joint, the locally different deformation behaviourhas to be taken into account. It is also published that different mechanical properties inthe heat-affected zone (HAZ) [5] as well as a weld metal with a lower strength as thebase metal [6] can be the reason or the starting point of a fracture in welded joints. Anew investigation demonstrates [7] that in TIG-welded joints of the high strength steelStE 690, a fine-grained area in the heat-affected zone with a lower strength than that ofthe base metal is exclusively the starting zone of fracture under cyclic loading in thefully compressive range. These investigations support the approach described here thatthe mechanical behaviour of the different kinds of microstructure in the heat-affectedzone of welded joints has to be taken into account in the deformation analysis. The in-fluences of these inhomogeneities on the local deformation behaviour of welded jointswere determined by experiments and numerical calculations over a wide range of tem-perature and loading. The numerical deformation analysis was performed with themethod of Finite Elements, in which recently developed deformation models simulatethe mechanical behaviour of materials over the tested range of temperature and loadingconditions.

The starting point of these investigations was the question if such deformationmodels are able to describe the deformation behaviour of welded joints sufficiently.

361

* Technische Universitat Braunschweig, Institut fur Schweißtechnik und Werkstofftechnologie,Langer Kamp 8, D-38106 Braunschweig, Germany

** Volkswagen AG, D-38436 Wolfsburg, Germany



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 themicrostructure in the normalized state consists of ferrite and minor amounts of bainiteand pearlite. The hardness has a value of 220 HV. The chemical composition is listedin Table 16.1.

For the deformation analysis, manual arc weldings were manufactured with twodifferent widths of the weld seam (24 mm and 16 mm) using the same welding param-eters – 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 weldedplates. The chemical composition of the electrodes is also given in Table 16.1 and thewelding parameters (Uw =22.5 V, Iw =170 A, vw =0.2 cm/s) lead to a heat input perunit 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 thethree 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 thesedifferent zones, but also to establish Finite-Element models for the calculations. Detailedexperimental and numerical investigations indicated that the heat-affected zone must alsobe divided into zones because the mechanical properties are not constant over its width.On the basis of the microsections, four significantly different kinds of microstructurecould be identified. The differences between these kinds of microstructure are causedby the peak temperature and the number of weld cycles. To gain the mechanical proper-ties of each microstructure, it must be identified and then prepared in specimens with alarge diameter and a large measurement length by using the so-called weld simulation.

During the weld simulation, various specimens of the base metal were conduc-tively heated up to different peak temperatures and then cooled under nitrogen with dif-ferent t8,5-cooling times. The simulation parameters for each structure can be deter-mined first of all numerically by using the thermal conduction equation and subse-quently optimized experimentally by comparison with microsections. The specimenswith homogeneously simulated microstructures over a measurement length of 25 up to30 mm were used in tensile tests, creep tests and tension-compression tests. Micro-graphs of all four kinds of microstructure are shown in Figure 16.3. The various micro-structures are listed in Table 16.2 together with their hardness values.

16 Consideration of Inhomogeneities in the Application of Deformation Models

362

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% Nb, V, S

Tenacito 70 0.06% 0.5% 1.6% – – – 0.9%

In addition to the investigations with the four kinds of microstructure of the heat-af-fected zone, the same mechanical tests were carried out with the base metal and theweld metal. The specimens containing the weld metal were taken from welded jointsvertical to the weld seam. They were machined in that way that in the mechanical tests,the deformation is concentrated in the weld metal.

16.2 Materials and Numerical Methods

363

Figure 16.1: Microsections of manual arc-welded joints of the steel StE 460; up: width of theweld 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).


364

Figure 16.3: Microstructure of the base metal and microstructures N, F, C and M in the heat-af-fected zone (from left to right).

Table 16.2: Kinds of investigated microstructures and Vickers hardness.

State of material Microstructure Hardness Notice

Base metal Ferrite (bainite, pearlite) 220 HVMicrostructure N Ferrite (bainite, pearlite) 230 HV fine-grained as base metalMicrostructure F Ferrite (bainite, pearlite) 270 HV fine-grained as base metalMicrostructure C Bainite, martensite 280 HVMicrostructure M Martensite (bainite) 375 HVWeld metal – 220 HV

16.2.2 Deformation models and numerical methods

16.2.2.1 Deformation model of Gerdes

In these investigations, the high-temperature formulation of the deformation model ofGerdes [8] was used:

��in � C1 exp � Fh

RT

� � �� kin��0

� �n1

sinhbA�

h�� kin�RT

� �� 1�

��kin �H1E exp � �1 bA�h �kin sign ��eff�

RT

� ��in

� R1 exp � Fh

RT

� �sinh

bA�h �kin

RT

� �� 2�

A��h � dV1��in� � dV2�� dV3 � �3�

The model parameter �0, which is used for the stress standardization, was substitutedby the Young’s modulus. The time-dependency of the activation volume is describedby 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 anevolution algorithm to gain the model parameters for the calculations. The parametercalculations were performed here only phenomenologically for each temperature andeach kind of microstructure. Additionally, the different types of tests were simulatedseparately because the qualities of the model parameters became much better then, andthe fitting calculations needed even less time than the parameter estimations made forcommon tensile and creep tests.

16.3 Investigations with Homogeneous Structures

All investigations were carried out to prove whether deformation models are able to de-scribe the mechanical behaviour of the steel StE 460, of four significant kinds of micro-structure of this steel and of the weld metal of manual arc weldings. Tensile, creep andtension-compression tests were performed over a wide range of temperatures (roomtemperature up to 700 �C) and loading conditions in order to characterize the mechani-cal behaviour of each state of the base metal.


365

16.3.1 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. Thebase metal and the weld metal show a clearly visible yield strength, which is highly pro-nounced in the base-metal deformation, whereas a proof strength has to be attributed to thevarious kinds of microstructure produced through weld simulation. The arrangement ofthe stress-strain curves in Figure 16.4 corresponds with the hardness values. The micro-structure M with the highest hardness values has the highest flow stresses. For the param-eter estimation, the stress-strain curve of the base metal has to be filtered so that the yieldstrength is transformed into a proof strength. The fitting calculations with the deformationmodel of Gerdes indicate that the mechanical behaviour of the various kinds of micro-structure and the weld metal can be described sufficiently well, but differences arise inthe simulation of the stress-strain curve of the base metal and its yield strength cannotbe simulated by the deformation model. The largest differences occur in the yield-strength range, where the stress values are underestimated. At large strains (≥3%), theexperimental and the calculated values differ less. The calculated stress-strain curve ofthe high strength microstructure M shows a stress state of saturation, whereas a steady-strain hardening is observed in the experiments with this kind of microstructure.

The tensile tests carried out at 300 �C show very similar results as the tests atroom temperature (Figure 16.5). The arrangement of the stress-strain curves has thesame order and corresponds also to hardness values. The mechanical behaviour of thebase metal differs from that at room temperature because the yield strength of the base


366

Figure 16.4: Stress-strain curves of the investigated microstructures at room temperature; sym-bols: experimental curve; lines: fitted curve.

metal is not as pronounced as at room temperature. Comparing Figure 16.4 and Figure16.5, one sees that at strains above 3%, the strength values of all microstructures arehigher at 300 �C than at room temperature. The Young’s modulus of all microstructuresdecreased to a value of 170 000 MPa. The mechanical behaviour of all kinds of micro-structure can be described again sufficiently by the fitting calculations and here, theyield strength of the base metal is also adequately simulated. Only the calculatedstress-strain curve of the microstructure M includes a stress state of saturation at largestrains although the real stress-strain curve exhibits a strain-hardening behaviour.

The stress-strain curves at 500 �C (Figure 16.6) are again arranged in the same orderas at room temperature. At this temperature, the base metal shows no yield-strength effectsand its stress-strain curve is nearly the same as the curve of the microstructure N. Thestress-strain curve of the microstructure M reveals a softening behaviour at largestrains, 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 150000 MPa simulatethe mechanical behaviour of all kinds of microstructure very well, only the softening of themicrostructure 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 �C are notarranged in the same order as at room temperature. At 700 �C, the base metal has ahigher strength than the microstructures N and F because the especially low grain sizeof these two microstructures favours plastic deformation at high temperatures. The twohigh strength microstructures M and C show a softening behaviour caused by a trans-formation of the microstructure. The fitting calculations (Young’s modu-lus=130 000 MPa) correspond relatively well with the experimental results although thesoftening behaviour is unsuitably modelled by horizontal lines.


367

Figure 16.5: Stress-strain curves of the investigated microstructures at 300 �C; symbols: experi-mental curve; lines: fitted curve.


368



16.3.1.2 Creep tests

The creep tests were carried out at 500 �C and 700 �C. Already at 500 �C (loading:275 MPa), all kinds of microstructure show a remarkable creep (Figure 16.8). The or-der of all creep curves agrees within the range of reproducibility with the results of thetensile tests. The differences between the base metal and the microstructure N and thedifferences between the microstructures M and C are very small. The fitting calcula-tions show a very good applicability of the deformation model of Gerdes to the creepbehaviour of all microstructures. Differences between the experiments and the calcula-tions are not noticeable in Figure 16.8.

The analysis of the creep behaviour at 700 �C (loading: 50 MPa) reveals resultsanalogous to those of the tensile tests. The highest creep strains occur in the micro-structures F and N (Figure 16.9), whereas the smallest creep strains occur in the micro-structures C and M. The base metal takes a middle position of all creep curves. The be-haviour of the microstructures F and N is highly influenced by the fine-grained micro-structure, which favours the plastic deformation. The fitting calculations simulate thecreep behaviour sufficiently well. The softening behaviour of the two microstructuresM and C caused by a transformation of the microstructure is modelled as steady-statecreep. It must be stated that it cannot be anticipated to find the consequences of thetransformation in the results because this behaviour has not been implemented in themodel equations.


369

Figure 16.8: Creep curves of the investigated microstructures at 500 �C, loading: 275 MPa; sym-bols: experimental curve; lines: fitted curve.

16.3.1.3 Cyclic tension-compression tests

Strain-controlled cyclic tension-compression tests were performed with varying strainamplitudes at room temperature (±0.6%) and at 500 �C (±0.4%). The strain rates of thetests lie between 1 · 10–4 1/s and 5 ·10–4 1/s. The fitting calculations were made in twosteps because of the hardening or softening behaviour found in the different kinds ofmicrostructure. In the first step, the first five cycles of the tension-compression testswere used to estimate the first model parameters. If necessary, these model parameterswere utilized in the second step as the starting set of parameters for new parameter esti-mations to simulate the behaviour during the whole tension-compression tests.

Figure 16.10 shows the results of the first five cycles of all microstructures atroom temperature (strain rate: �� 5 � 10�4 l/s). The order of all curves is the same asthat of the stress-strain curves achieved from tensile tests. The base metal exhibits apronounced yield strength, which has to be filtered for the fitting calculations. Themathematical simulations describe the mechanical behaviour of all microstructures suf-ficiently although the yield strength cannot be modelled by the deformation model. Thecyclic hardening of all microstructures can be described by the model.

Figure 16.11 contains the stress-time curves of all microstructures over the com-plete testing time. The printed symbols are the points of return in the fully tensionrange. It can be seen that all microstructures with the exception of the base metal soft-en after the first hardening cycles. The hardening period lasts about 5 cycles and thesoftening leads very fast (10 to 20 cycles) to a state of cyclic saturation. Only the basemetal already softens from the beginning of the test, and the cyclic state of saturationis reached after less cycles.


370

Figure 16.9: Creep curves of the investigated microstructures at 700 �C, loading: 50 MPa; sym-bols: experimental curve; lines: fitted curve.


371

Figure 16.10: Stress-time curves of the investigated microstructures at room temperature, first fivecycles, strain amplitude: ±0.6%; symbols: experimental curve; lines: fitted curve.

Figure 16.11: Stress-time curves of the investigated microstructures at room temperature, 200 cy-cles, strain amplitude: ±0.6%; symbols: experimental curve; lines: fitted curve.

In Figure 16.12, a photo taken with the transmission electron microscope showsthe saturated state of the base metal. The state of saturation can be identified throughthe array of dislocations in a cell structure. The fitting calculations made in the secondstep indicate that the deformation model of Gerdes is not able to simulate the cyclicsoftening behaviour of all microstructures. Figure 16.11 demonstrates that the stressesin the initial hardening range were underestimated and in the following softening rangeoverestimated. The model describes the state of saturation by a horizontal line for allmicrostructures.

The results of cyclic tension-compression tests carried out at 500 �C are repre-sented in the following figures (strain rate: �� 5 � 10�4 l/s). Figure 16.13 contains thefirst five cycles of the tension-compression tests of all microstructures. The curves arearranged in the same order as the curves of the tensile tests.

The cyclic behaviour agrees with the results found at room temperature. It can beseen that all microstructures with exception of the base metal firstly harden and thensoften during the cyclic loading (Figure 16.14). The amounts of hardening and soften-ing are not as high as at room temperature. The fitting calculations simulated the firstfive cycles of the stress-strain behaviour sufficiently (Figure 16.13). Because of the re-latively weak softening behaviour, the model parameters achieved from the first fivecycles could also be used to describe the behaviour during all following cycles. A sec-ond 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 ofcyclic saturation. The cyclic stress-time curves of all microstructures can be simulatedsufficiently well by the deformation model. But, as to be seen, the cyclic softeningafter the cyclic hardening cannot be described by the model. The stresses are underesti-mated in the range of hardening and overestimated in the range of softening, but thedifferences between the experiments and calculations remain relatively small.

16.3.2 Discussion

All results indicate that recently developed deformation models can be successfully ap-plied to describe the mechanical behaviour of a microalloyed steel and of the differentmicrostructures identified in the heat-affected zone of its weldings. But, in view of ac-curacy and calculation time, it is useful for all fitting calculations to estimate the pa-rameters separately for each microstructure, each temperature and each testing se-quence. The fitting calculations are carried out phenomenologically as it was not possi-ble to find relations between model parameters and microstructural parameters. Somespecial aspects of the mechanical behaviour of the microalloyed steel and its variouskinds 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 thisrange are underestimated and the model calculates only a proof strength. Secondly, thesoftening behaviour of some kinds of microstructure at high temperatures cannot be de-scribed with the deformation model. The model simulates the softening behaviourthrough a steady-state behaviour. This behaviour is also detectable in the simulation ofcyclic tension-compression tests. In this case, the hardening and softening behaviour is


372


373

Figure 16.12: TEM-photo of the state of saturation due to cyclic loading, base metal at room tem-perature after 200 cycles.

Figure 16.13: Stress-time curves of the investigated microstructures at 500 �C, first five cycles,strain amplitude: ±0.4%; symbols: experimental curve; lines: fitted curve.

also modelled by a steady-state or saturation behaviour. The experimentally observedsaturation behaviour of all microstructures is modelled sufficiently well.

From these results, a simplification for the Finite-Element calculations of the de-formation behaviour of welded joints can be derived. In order to lower the calculationtime, the number of zones in the heat-affected zone can be reduced if the mechanicalbehaviour 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 ofwelded 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 load-ing, and calculations were performed with the Finite-Element code ABAQUS with thesame control of the strain rate as in the experiments.


374

Figure 16.14: Stress-time curves of the investigated microstructures at 500 �C, 200 cycles, strainamplitude: ±0.4%; symbols: experimental curve; lines: fitted curve.

16.4.1 Deformation behaviour of welded joints

16.4.1.1 Experimental investigations

The experiments to gain the strain distributions were performed in cooperation with theproject C2. In these experiments, flat specimens taken from the welded joints verticalto the weld seam were tested in tensile tests with the same strain control as in the testswith the homogeneous structures. The strain distributions were determined with thegrating method [9]. For reasons of symmetry and of the grating size, only less than onehalf of the welded specimens was observed during the tests. The inspected regionswere the heat-affected zone, the weld metal and small parts of the base metal. Thewelded specimens show relatively large rigid body motions during the tests so that ref-erence objects had to be fixed to the specimens. These reference objects are subject tothe same rigid body motions as the welded specimens, but they are not deformed dur-ing 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 incooperation 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 distri-butions and microsections. It turned out during the investigations that only those pointshad to be determined describing the transition from the base metal to the heat-affectedzone and from the heat-affected zone to the weld seam (see Figure 16.2). Then, themesh for the heat-affected zone was modelled with four equidistant zones containingthe information of the mechanical behaviour of the affiliated microstructures. The Fi-nite-Element calculations were performed with the model parameters gained from theroom-temperature fittings. After the first experiments, it could be observed that the de-formations are concentrated in the weld metal so that it should be possible to modelthe heat-affected zone with one microstructure only. In some calculations, the heat-af-fected zone was modelled only with the microstructure C. This procedure reduces thecalculation times for the tensile test simulation with ABAQUS. Therefore, the follow-ing sections include calculations for 6-material models (weld metal, four regions of theHAZ, 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 realtensile tests was used. The load was attached as a boundary condition on one side of


375

the Finite-Element mesh. The first calculations revealed that the calculation times arevery high in comparison to the real tests. The cpu-time to calculate a tensile test of 900seconds was about 20000 cpu-seconds with the 6-material model (2300 elements) andabout 4000 cpu-seconds with the 3-material model (900 elements). A serious problemfor the Finite-Element calculations is the time-stepping during the first part of the simu-lation (0 to 40 seconds). During this period, the time-steps for the numerical integrationof the model equations are reduced from 5 seconds to 0.05 seconds because the differ-ences between the material properties of the weld seam and the microstructures M or Care 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 specimenswith a weld metal length of about 24 mm. The analysis of the measured strain distribu-tions during loading shows the following results. The first remarkable strains occur inthe weld metal possessing the lowest flow stresses of all microstructures. The deforma-tions in the heat-affected zone and in the base metal are smaller. Figure 16.15 illus-trates the distributions of the longitudinal strains measured with the grating method at astage of 2.2% medium strain of the whole specimen. Only the strains in the weld-metalzone and in the heat-affected zone are visible because the grating was fixed only onthese zones. The longitudinal strains have maximum values of about 4.8% in the weld-metal and less than 1% in the heat-affected zone. The curvature of the strain isolines in-dicates that the soft weld metal is backed up by the harder heat-affected zone. The hin-dered vertical deformations in the transition zone between weld-metal and heat-affectedzone influence not only the vertical strains (Figure 16.15 right) but also the distributionof the longitudinal strains (Figure 16.15 left).

The strain calculations made with a 6-material model show nearly the same re-sults (Figure 16.16) as the experiments, but there are also some differences. The firstremarkable strains occur in the heat-affected zone in the region of the microstructuresN and F because they have a proof strength lower than the yield strength of the basemetal. But at a medium strain stage of 2.2%, the longitudinal strains in the weld metalare much higher than the strains in the heat-affected zone. The calculated values reach3.4% in the weld-metal and less than 1% in the heat-affected zone. The strains in thebase metal are already higher than in the heat-affected zone but smaller than in theweld metal. The backing up of the soft weld metal by the harder heat-affected zone isalso determined. At the points, where the contact faces between weld-metal and heat-af-fected zone break through the free surfaces, singularities appear in the calculated stressand strain distributions. These singularities are caused by the sudden change of the ma-terial properties between two neighbouring elements of the weld-metal and the heat-af-fected 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 experi-mental results. Only the strain distributions in a wider range around the singularitiescan be compared with experimental results.


376


377

Figure 16.15: Strain distributions of a welded specimen with a long weld-metal zone measuredwith 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 calculatedwith the 6-material model, medium strain: 2.2%; left: longitudinal strains; right: vertical strains.

The singularities also occur in the analysis of the strains calculated with a 3-material model. Here, the first remarkable strains do not occur in the heat-affected zonebut in the weld metal as observed in the experiments. The strain values at a mediumstrain stage of 2.2% are about 3.5% in the weld-metal and less than 1% in the heat-af-fected zone (Figure 16.17). The backing up of the soft weld metal is also determined inthese investigations.

The next examined load step at a medium strain of 4.3% was at the end of thetensile test. Figure 16.18 illustrates the strain distributions of this state. The curvaturesof the strains are the same as in the load step discussed before. They also demonstratethe backing up of the soft zone by a harder zone. Figure 16.18 shows the longitudinalstrains, which have values of more than 8% in the weld-metal and less than 2% in theheat-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 thespecimen than at the free surface.

The calculations for a load step of 4.5% show qualitatively the same results asthe experiments. The strains calculated with the 6-material model and the 3-materialmodel are represented in Figures 16.19 and 16.20.

In both figures, the calculated strains in the weld metal are lower than the mea-sured strains. Both maximum values of the longitudinal strains in the weld metal areslightly lower than 7%. The arrangement of the strain isolines in the figures corre-sponds with the experimental results and confirms the backing up of the soft weld met-al by the harder microstructure of the heat-affected zone. The figures representing thevertical 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.


378



379

Figure 16.18: Strain distributions of a welded specimen with a long weld-metal zone measured bythe grating method, medium strain: 4.3%; 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 zoneof weld metal are shown in Figure 16.21. At the end of the tensile tests (medium strain4.3%), the longitudinal strains have values of about 9% in the weld-metal and lowerthan 2% in the heat-affected zone (Figure 16.21). The strains in the base metal aresomewhat higher than 2%. The vertical strains in Figure 16.21 indicate the backing upof the soft weld metal and base metal through the hard heat-affected zone. The curva-ture of the strain isolines demonstrates this effect clearly. It can be seen that the strainsmeasured in specimens of a welded joint with a narrow weld seam are slightly higherthan 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 the3-material model (Figure 16.23) correspond with the experimental results. The backingup of the weld metal is also confirmed by the longitudinal and vertical strain distribu-tions. But the calculated strain values in the weld metal are lower than the measuredones. The maximum strains in the weld metal have values of 7%, and the strains in theheat-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 modelsare able to describe the mechanical behaviour of welded joints sufficiently well. Prob-lems arise in the numerical investigations with the automatic time-stepping in the Fi-


380



381

Figure 16.21: Strain distributions of a welded specimen with a short weld-metal zone measuredwith 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 calculatedwith the 6-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 neigh-bouring materials in the transition range between the weld metal and the hard micro-structure in the heat-affected zone. Also, the calculated strain and stress singularitiesare not found in the experiments, and they are too caused by the sudden change of thematerial properties. The 3-material model and the 6-material model lead to nearly thesame calculated strains so that the number of different regions in the heat-affected zoneof 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 towelded joints. The mechanical behaviour of the base metal and of various kinds of mi-crostructure is in a reasonable good accordance with the experimental results.

But the analysis demonstrates also that some special aspects of the mechanical be-haviour of microalloyed steels cannot be simulated by the deformation model. Thislack of simulation includes yield-strength effects and the softening behaviour, which oc-curs in the high temperature and cyclic range. New deformation models have to de-scribe these effects if they should be applied in further investigations. A big problem inthe calculations was the handling of the equations in Finite-Element calculations and in


382

Figure 16.23: Strain distributions of a welded specimen with a short weld-metal zone calculatedwith the 3-material model, medium strain: 4.5%; left: longitudinal strains; right: vertical strains.

the parameter estimation. Both numerical methods need very high calculation times sothat numerical investigations can only be made if the hardware is big enough. Furtherinvestigations may solve this problem if numerical methods or deformation equationsare developed shortening the calculation times by faster algorithms or if model equa-tions are established, which are easier to handle in modern calculation methods.

The efforts of modelling welded joints can also be reduced according to these in-vestigations. If the highest strains or fractures do not occur in the heat-affected zone, itmust not be modelled with more than one microstructure. In this case, the Finite-Ele-ment meshes can directly be derived from hardness distributions of welded joints. Inother cases, it can be recommended from these investigations that only two differentmicrostructures have to be taken into account for the calculation of the mechanical be-haviour of welded joints.

References

[1] U.H. Clormann: Örtliche Beanspruchungen von Schweißverbindungen als Grundlage desSchwingfestigkeitsnachweises. Dissertation, Technische Hochschule Darmstadt, 1986.

[2] W. Schieblich: Rechnerische und experimentelle Ermittlung des Zeitstandverhaltens eineraustenitischen Schweißverbindung. Dissertation, Technische Hochschule Darmstadt, 1992.

[3] W. Eckert: Experimentelle und numerische Untersuchungen zum Zeitstandverhalten vonSchweißverbindungen der Werkstoffe X20 CrMoV 12 1 und GS-17 CrMoV 5 11. Dissertation,Staatliche Materialprufanstalt (MPA) Stuttgart, 1992.

[4] M. Kaffka: Beitrag zum Zeitstandverhalten artgleicher Schweißverbindungen einer Nickelba-sislegierung unter besonderer Berucksichtigung des lokalen Verformungsverhaltens. Disserta-tion, Technische Hochschule Aachen, 1985.

[5] U. Patzold: Verformungsanalyse von Schweißverbindungen. Dissertation, Technische Univer-sitat Braunschweig, 1992.

[6] H. Hickel: Eigenspannungen und Festigkeitsverhalten von Schweißverbindungen. Disserta-tion, Universitat Karlsruhe, 1973.

[7] J. Pucelik, Th. Nitschke-Pagel, H. Wohlfahrt: Relationship of tensile residual stresses and fa-tigue crack propagation under cyclic loading in the fully compressive range. Poster at theFourth European Conference on Residual Stresses, 4.–6. 06. 1996, Cluny (F), to be pub-lished.

[8] R. Gerdes: Ein stochastisches Werkstoffmodell fur das inelastische Materialverhalten metal-lischer Werkstoffe im Hoch- und Tieftemperaturbereich. Dissertation, Technische UniversitatBraunschweig, 1995.

[9] D. Winter: Optische Verschiebungsmessung nach dem Objektrasterprinzip mit Hilfe eines fla-chenorientierten Ansatzes. Dissertation, Technische Universitat Braunschweig, 1993.

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Bibliography

Theses Resulted from the Collaborative Research Centre

1985–1987

Subproject A2D. Rode: Ermudungsverhalten anisotroper Aluminiumlegierungen unter mehrachsigen Beanspru-

chungen im Zeitfestigkeitsbereich.

Subproject A3C. Schulze: Optische Verformungsmessungen an Schalen nach dem Rasterprinzip.

Subproject A6I. Gobel: Modellbildung fur die Hochtemperaturplastizitat mit Hilfe metallphysikalischer Ergeb-

nisse.T. Losche: Entwicklung eines Stoffgesetzes fur die Hochtemperaturplastizitat auf der Grundlage

von Markov-Prozessen.R. Schettler-Kohler: Entwicklung eines makroskopischen Stoffgesetzes fur Metalle aus einem sto-

chastischen Zwischenmodell.G. Wilhelms: Ein phanomenologisches Werkstoffgesetz zur Beschreibung von Plastizitat und Krie-

chen metallischer Werkstoffe.

Subproject B4W. Kohler: Beitrag zur Wasserstoffversprodung metallischer Werkstoffe im LCF-Bereich.

Subproject D1R. Linnemann: Beitrag zur Bewertung von Schweißnahtfehlern mittels bruchmechanischer Metho-

den.

1988–1990

Subproject A1U. Meyer: Einfluss verschiedener Verformungsparameter auf die Rekristallisationskinetik von

Kupfer.J. Schmidt: Untersuchung der Erholungs- und Rekristallisationsvorgange in tieftemperaturver-

formten Metallen mit Hilfe kalorischer Messungen.

Subproject A5/B4R. Schubert: Verformungsverhalten und Risswachstum bei Low-Cycle-Fatigue.

Subproject B2M. Schwesig: Inelastisches Verhalten metallischer Werkstoffe bei hoheren Temperaturen – Nume-

rik und Anwendung.

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Subproject B5H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Berucksichtigung in

Konstruktionsberechnungen.E. Beißner: Zum Tragverhalten stahlerner Augenstabe im elastisch-plastischen Zustand.

Subproject C3K. Wolf: Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen aus

kohlenstoffaserverstarktem Kunststoff.

1991–1993

Subproject A5/B4H. Klingelhoffer: Rissfortschritt und Lebensdauer bei Hochtemperatur Low-Cycle-Fatigue in kor-

rosiven Gasen.

Subproject A6/B1H. Hesselbarth: Simulation von Versetzungsstrukturbildung, Rekristallisation und Kriechschadi-

gung mit dem Prinzip der zellularen Automaten.F. Kublik: Vergleich zweier Werkstoffmodelle bei ein- und mehrachsiger Versuchsfuhrung im

Hochtemperaturbereich.H. Schlums: Ein stochastisches Werkstoffmodell zur Beschreibung von Kriechen und zyklischem

Verhalten metallischer Werkstoffe.H. Grohlich: Finite-Element-Formulierung fur vereinheitlichte inelastische Werkstoffmodelle ohne

explizite Fließflachenformulierung.

Subproject A8R. Neuhaus: Entwicklung der Versetzungsstruktur und Hartung bei [100] und [111] orientierten

Kupfer und Kupfer-Mangan Kristallen im einachsigen Zugversuch.A. Kalk: Dynamische Reckalterung und die Grenzen der Stabilitat plastischer Verformung. Sys-

tematische Untersuchungen an Viel- und Einkristallen aus CuMn.

Subproject A9A. Hampel: Struktur und Kinetik der lokalisierten Verformungen in kubisch flachenzentrierten Le-

gierungseinkristallen – Experimente und Modellrechnungen.

Subproject A10M. Muller: Plastische Anisotropie polykristalliner Materialien als Folge der Texturentwicklung.

Subproject B2G. Kracht: Erschließung viskoplastischer Stoffmodelle fur thermomechanische Strukturanalyse.E.-R. Tirpitz: Elasto-plastische Erweiterung von viskoplastischen Stoffmodellen fur Metalle.H. Braasch: Ein Konzept zur Fortentwicklung und Anwendung viskoplastischer Werkstoffmodelle.

Subproject B5/B7M.M. El-Ghandour: Low-Cycle Fatigue Damage Accumulation of Structural Steel St52.

Subproject B6G. Zhang: Einspielen und dessen numerische Behandlung von Flachentragwerken aus ideal plas-

tischem bzw. kinematisch verfestigendem Material.

Subproject B8R. Mahnken: Duale Methoden fur nichtlineare Optimierungsprobleme.

Subproject C2W. Wilke: Photogrammetrische Verformungsmessung durch Überlagerungseffekte frequenzmodu-

lierter periodischer Bildstrukturen.

Theses Resulted from the Collaborative Research Centre

385

J.O. Hilbig: Zur Beeinflussung der Speckleauswertung durch eine Kombination der Speckle- undder Objektrastermethode.

D. Winter: Optische Verschiebungsmessung nach dem Objektprinzip mit Hilfe eines flachenorien-tierten Ansatzes.

Subproject C3P. Horst: Zum Beulverhalten dunner, bis in den plastischen Bereich zyklisch durch Schub belas-

teter Aluminiumplatten.D. Hachenberg: Untersuchungen zum Nachbeulverhalten stringerverstarkter schubbeanspruchter

Platten aus kohlenstoffaserverstarktem Kunststoff.

Subproject C4U. Patzold: Verformungsanalyse von Schweißverbindungen.

Publications Resulted from the Collaborative Research Centre

1985–1987

Subproject A1W. Witzel, F. Haeßner: Zur Vergleichbarkeit von Werkstoffzustanden nach Dehnen, Stauchen und

Tordieren. Z. Metallkunde 78 (1987) 316.F. Haeßner, H. Muller: Einfluss von Korngroße und Beanspruchungsrichtung auf die Fließspan-

nung texturbehafteter Zink-Bleche. Z. Metallkunde 78 (1987) Juli-Heft.

Subproject A2G. Lange, D. Rode: Ermudungsverhalten anisotroper Aluminium-Legierungen unter mehrachsiger

Beanspruchung im Zeitfestigkeitsbereich. Hauptversammlung der Deutschen Gesellschaft furMetallkunde, 20.–23. Mai 1986, Vortrag Nr. EBO7.

D. Rode: Ermudungsverhalten anisotroper Aluminium-Legierungen unter mehrachsiger Beanspru-chung im Zeitfestigkeitsbereich. Dissertation TU Braunschweig, 1987.

Subproject A3K. 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 Krummungsermittlung mit Hilfe des Phasenshiftprinzips.

Technisches Messen 6 (1987).S. Angerer, J.P. Lowenau, B. Morche, W. Wilke: 3D-Verformungsmessung an einem gummiarti-

gen Dampfungselement mit Hilfe der Stereophotographie und der digitalen Bildverarbeitung.Fachkolloquium Experimentelle Mechanik, Stuttgart, 1986, pp. 33–40.

H.C. Gotting, R. Schutze, R. Ritter, W. Wilke: Dehnungsmessungen an Faserverbundwerkstoffenmit Hilfe des Beugungsprinzips. VDI-Berichte 631, VDI Verlag Dusseldorf, 1987, pp. 275–285.

J. Hilbig, R. Ritter: Zur Bestimmung von Neigungsanderungen schalenformiger Objekte mit Hilfeder 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. ofthe 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 experi-mentellen Spannungsanalyse, Dusseldorf, 1987.

R. Ritter, W. Wilke: Dehnungsmessung nach dem Gitterprinzip. Lecture at the Colloquy of theCollaborative Research Centre (SFB 319), Goslar, 4.–5. Dezember 1986.

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C. Schulze: Optische Verformungsmessung an Schalen nach dem Rasterprinzip. Dissertation ander Fakultat fur Maschinenbau und Elektrotechnik der TU Braunschweig, 1986.

Subproject A5K.-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 Interac-tion. Fifth International Conference on Mechanical Behaviour of Materials, Peking, 1987.

K.-T. Rie, R.-M. Schmidt: Lifetime Prediction under Creep-Fatigue Conditions. Proceedings ofthe Second International Conference on Low-Cycle Fatigue and Elasto-Plastic Behaviour ofMaterials, September 1987, Munchen.

R.-M. Schmidt: Low-Cycle Fatigue under Special Aspects of Quality Control. Braunschweig Kol-loquium 1985, DVS, pp. 107–119.

K.-T. Rie, R.-M. Schmidt: High Temperature Low-Cycle Fatigue of Austenitic and Ferritic Weld-ments. ECF 6, Amsterdam, 1986, pp. 1096–1113.

K.-T. Rie, R.-M. Schmidt: Ermudung von Schweißverbindungen im Bereich geringer Schwing-spielzahlen. 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, B1E. Steck: Entwicklung von Stoffgesetzen fur die Hochtemperaturplastizitat. Grundlagen der Um-

formtechnik. 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 Plas-

ticity 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 fur die Wechselwirkung von Plastizitat und Kriechen. Work-

shop Werkstoff und Umformung, Stuttgart 1986, Springer Verlag, 1986.

Subproject A8Th. Wille, W. Gieseke, Ch. Schwink: Quantitative analysis of solution hardening in selected cop-

per 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 A9H. Neuhauser, 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. Neuhauser: Physical manifestation of instabilities in plastic flow. In: V. Balakrishnan, C.E.

Bottani (Eds.): Mechanical Properties and Behaviour of Solids: Plastic Instabilities World Sci-entific, Singapore, 1986, pp. 209–252.

O.B. Arkan, H. Neuhauser: Dislocation velocities in Cu-Ni alloys determined by the stress pulse-etch pit technique and by slip line cinematography. phys. stat. sol. (a) 99 (1987) 385–397.

H. Neuhauser, O.B. Arkan: Dislocation motion and multiplication in Cu-Ni single crystals. phys.stat. sol. (a) 100 (1987).

Subproject B2H.K. Nipp: Temperatureinflusse auf rheologische Spannungszustande im Salzgebirge. Report Nr.

82-36 from the Institut fur Statik der TU Braunschweig.A. Schmidt: Berechnung rheologischer Zustande im Salzgebirge mit vertikalen Abbauen in Anleh-

nung an In-situ-Messungen. Report Nr. 84-43 from the Institut fur Statik der TU Braun-schweig.


387

Subproject B4K.-T. Rie, R. Schubert: Einfluss einer Druckwasserstoffumgebung auf das Ermudungs- und Riss-

wachstumsverhalten bei Low-Cycle Fatigue; Wasserstoff in Metallen. Results of the Schwer-punktprogramm, Deutsche Forschungsgemeinschaft, Bonn, 1986.

K.-T. Rie, R. Schubert: Note on the Crack Closure Phenomenon in LCF. 2. Int. Conf. on LCF andElasto-Plastic Behaviour of Materials, Munchen 1987, Elsevier Applied Science, pp. 575–580.

Subproject C1K. 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 theCollaborative Research Centre (SFB 319), Braunschweig.

K. Andresen, F. Hecker: Das Phasenshiftverfahren zur digitalen Auswertung von Moire-Musternund 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. Lowenau, B. Morche, W. Wilke: 3D-Verformungsmessung an einem gummiarti-gen Dampfungselement 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 CollaborativeResearch Centre (SFB 319), Goslar.

K. Andresen: Verformungsmessung mit Rasterverfahren und der digitalen Bildverarbeitung. Ober-seminar Mechanik, Universitat Hannover.

K. Andresen: Konturermittlung und Verformungsmessung mit der digitalen Bildverarbeitung. Me-chanik/Colloquy of the Collaborative Research Centre (SFB), TU Berlin.

Subproject D1J. Ruge, R. Linnemann: Festigkeits- und Verformungsverhalten von Bau-, Beton- und Spannstah-

len bei hohen Temperaturen. Arbeitsbericht 1984–1986. Subproject B4, Collaborative ResearchCentre (SFB 148), TU Braunschweig.

R. Linnemann: Beitrag zur Bewertung von Schweißnahtfehlern mittels bruchmechanischer Metho-den. Dissertation TU Braunschweig, 1987.

Subproject D2G. Bahr: Kommunizierende Versuchstechnik und simultane theoretische und experimentelle Trag-

werksuntersuchung. Dissertation TU Braunschweig, 1984.W. Maier, M. Klahold: Das Konzept der kommunizierenden Versuchstechnik fur Stabtragwerke.

Nachdruck eines Vortrages auf dem Fach-Kolloquium Experimentelle Mechanik 1986, Eigen-verlag, Ingenieurwiss. Zentrum, Inst. fur Modellstatik, Uni Stuttgart.

1988–1990

Subproject A1F. Haeßner, K. Sztwiertnia: The Misorientation Distributions Associated with the Texture of Poly-

crystalline 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 alumi-nium determined with a low temperature calorimeter. Scripta Met. 22 (1988) 1917.

U. Meyer: Einfluss verschiedener Verformungsparameter auf die Rekristallisationskinetik vonKupfer. Dissertation TU Braunschweig, 1989.

Bibliography

388

J. Schmidt: Untersuchung der Erholungs- und Rekristallisationsvorgange in tieftemperaturver-formten Metallen mit Hilfe kalorischer Messungen. Dissertation TU Braunschweig, 1990.

J. Schmidt, F. Haeßner: Stage III-Recovery of Cold Worked High-Purity Aluminium Determinedwith 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.): Recrys-tallization ’90, The Minerals, Metals and Materials Society, 1990, pp. 511.

Subproject A2D. Rode, G. Lange: Beitrag zum Ermudungsverhalten mehrachsig beanspruchter Aluminiumle-

gierungen. Metall 42 (1988) 582.

Subproject A5/B4R. Schubert, K.-T. Rie: Verfestigung, Fließflache und Versetzungsstruktur bei Low-Cycle-Fatigue.

Mat.-wiss. und Werkstofftech. 19 (1988) 376–383.K.-T. Rie, R. Schubert, J. Olfe: Untersuchungen zur Oberflachenausbildung, Mikrostruktur und

Ermudungsschadigung im LCF-Bereich. DFG-Kolloquium: Schadensfruherkennung und Scha-densablauf bei metallischen Bauteilen, Darmstadt, Sept. 1989, Berichtsband DVM, Berlin,1989, pp. 55–62.

R. Schubert: Verformungsverhalten und Risswachstum bei Low-Cycle-Fatigue. Dissertation TUBraunschweig, 1989.

J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messungen von Dehnungsfeldern bei Hochtempera-tur-LCF. Zeitschrift fur Metallkunde 81(11) (1990) 783–789.

Subproject A6/B1E. Steck: Constitutive Laws for Strain-, Strainrate- and Temperature Sensitive Materials. Key-

note, Proceedings 2. Intern. Conf. on Technology of Plasticity (ICTP), Stuttgart, Springer Ver-lag, 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 seinem60. Geburtstag, Institut fur 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 Sympo-sium 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 Engi-neering 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 A8Ch. 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 Al-loys 1, 1988, pp. 313–319.


389

Subproject A9A. Hampel, H. Neuhauser: Investigation of slip line growth in fcc Cu alloys with high resolution

in time. phys. stat. sol. (a) 104 (1987) 171–181.H. Neuhauser: The dynamics of slip band formation in single crystals. Res. Mechanica 23 (1988)

113–135.H. Neuhauser: On some problems in plastic instabilities and strain localization. Rev. Phys. Appl.

23 (1988) 571–572.A. Hampel, O.B. Arkan, H. Neuhauser: Local shear rate in slip bands of CuZn and CuNi single

crystals. Rev. Phys. Appl. 23 (1988) 695.A. Hampel, H. Neuhauser: 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 DynamicBehaviour of Materials, DGM Verlag, 1988, pp. 845–851.

H. Neuhauser: 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., Aedermanns-dorf, 1988, pp. 407–415.

A. Hampel, M. Schulke, H. Neuhauser: Dynamic studies of slip line formation on single crystalsof 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. Neuhauser: Dislocation groups, multipoles, and friction stresses in �-CuZn alloys.phys. stat. sol. (a) 109 (1988) 149–160.

H. Neuhauser: Plastic instabilities and the deformation of metals. In: D. Walgraef, N.M. Gho-niem (Eds.): Proc. NATO Adv. Study Inst. on Patterns, Defects and Materials Instabilities,Kluwer Acad. Publ., Dordrecht, 1990, pp. 241–276.

H. Neuhauser, J. Plessing, M. Schulke: Portevin-LeChatelier effect and observation of slip bandgrowth in CuAl single crystals. J. Mech. Beh. Metals 2 (1990) 231–254.

Subproject A10D. Besdo: Finite-Element-Analyses of Strain-Space-Represented Elastic-Plastic Media Using Sim-

plified 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. Lec-ture at the 13. Umformtechnischen Kolloquium Hannover 14./15. Marz 1990, HFF-Bericht Nr.11 (Ed.: E. Doege), HFF Hannoversches Forschungsinstitut fur Fertigungsfragen E.V.

D. Besdo, L. Ostrowski: On the Creep-Ratchetting of AlMgSiO.5 at Elevated Temperature – Ex-perimental Investigations. Proc. of the Fourth IUTAM Symposium on Creep in Structures,Krakow, 10.–14. Sept. 1990.

Subproject B2M. Schwesig: Inelastisches Verhalten metallischer Werkstoffe bei hoheren Temperaturen – Nume-

rik und Anwendung. Report Nr. 89-57 from the Institut fur Statik der TU Braunschweig, 1989.M. Schwesig, H. Ahrens, H. Duddeck: Erfahrungen aus der Anwendung des inelastischen Stoff-

gesetzes nach Hart. Festschrift Richard Schardt, THD Schriftenreihe Wissenschaft und TechnikS1, Darmstadt, 1990.

M. Schwesig, H. Braasch, G. Kracht, H. Duddeck, H. Ahrens: Erfahrungen aus der Anwendunginelastischer Stoffgesetze bei hoheren Temperaturen. In: D. Besdo (Ed.): Numerische Metho-den der Plastomechanik, Tagungsband; Hannover, 1989.

D. Dinkler, M. Schwesig: Numerische Losung von Anfangswertproblemen in der Statik und Dyna-mik. Festschrift Heinz Duddeck, Braunschweig, 1988.

H. Duddeck, B. Kroplin, D. Dinkler, J. Hillmann, W. Wagenhuber: Berechnung des nichtlinearenTragverhaltens dunner Schalen im Vor- und Nachbeulbereich. Nichtlineare Berechnungen imKonstruktiven Ingenieurbau, Hannover, DFG-Gz.: Du 25/28-7, 1989.

D. Dinkler: Stabilitat elastischer Tragwerke mit nichtlinearem Verformungsverhalten bei instatio-naren Einwirkungen. Ingenieur-Archiv 60 (1989).

Bibliography

390

D. Dinkler: Stabilitat dunner Flachentragwerke bei zeitabhangigen Einwirkungen. Report Nr. 88-52 from the Institut fur Statik der TU Braunschweig, 1988.

H. Duddeck, D. Winselmann, F. T. Konig: Constitutive laws including kinematic hardening forclay 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 cohe-sive soil problems. Eurodyn 90, European Conference on Structural Dynamics, Bochum, DFG-Gz.: Du 25/34-1-2, 1990.

R. Meyer, H. Ahrens: An elastoplastic model for concrete. Conference Proceedings of the SecondWorld Congress on Computational Mechanics, Stuttgart, 1990.

Subproject B5H.-J. Scheibe: Zum zyklischen Materialverhalten von Baustahl und dessen Berucksichtigung in

Konstruktionsberechnungen. Dissertation TU Braunschweig, 1990.J. Scheer, H.-J. Scheibe, M. Reininghaus: Wirtschaftliche Bemessung von Schraubenanschlussen

bei Ausnutzung des duktilen Verhaltens von Stahl. Report Nr. 6018, Institut fur Stahlbau, TUBraunschweig, 1989.

J. Scheer, H.-J. Scheibe, D. Kuck: Untersuchungen von Tragerschwachungen unter wiederholterBelastung bis in den plastischen Bereich. Report Nr. 6099, Institut fur Stahlbau, TU Braun-schweig, 1989.

E. Beißner: Zum Tragverhalten stahlerner Augenstabe im elastisch-plastischen Zustand. Disserta-tion TU Braunschweig, 1989.

H.-J. Scheibe: Zur Berechnung zyklisch beanspruchter Stahlkonstruktionen im plastischen Be-reich. Lecture 8. Stahlbau-Seminar, Steinfurt, 1989.

Subproject B6R. 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. Konig: Micromechanical modelling and computation of

shake down with nonlinear kinematic hardening inducing examples for 2-D problems. Proc. ofCSME Mechanic Engineering Forum, Toronto, 1990.

E. Stein, G. Zhang, J.A. Konig: Micromechanic modelling of shake down with nonlinear kine-matic hardening inducing examples for 2-D problems. In: Axelrad, Muschik (Eds.): RecentDevelopments in Micromechanics, Springer Verlag, 1990.

Subproject C1K. Andresen, R. Helsch: Automatische Rasterkoordinatenermittlung mit Hilfe digitaler Filter. In-

formatik Fachberichte 149, Mustererkennung, 1987, pp. 228.K. Andresen, R. Helsch: Calculation of Grating Coordinates Using a Correlation Filter Tech-

nique. Optik 80 (1988) 76–79.K. Andresen, B. Kamp, R. Ritter: Verformungsmessungen an Rissspitzen nach dem Objekt-Raster-

Verfahren. VDI-Berichte 679 (1988) 393–403.K. Andresen, B. Morche: Die Ermittlung von Rasterkoordinaten und deren Genauigkeit. Muster-

erkennung 198, pp. 277–283. 10. DAGM-Symposium Zurich, 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-Bildauswer-

tung. 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 Werkstoffprufung und Qualitatskontrolle.

Tagungsband: Bildverarbeitung: Forschen, Entwickeln, Anwenden, Techn. Akad. Esslingen,1989.


391

K. Andresen, R. Helsch: Calculation of Analytical Elements in Space Using a Contour Algo-rithm. ISPRS-Commission V. Symp. on Close Range Photogrammetry and Machine Vision,Zurich. SPIE 1395 (1990) 863–869.

K. Andresen: Evaluation of Moire Fringes Using Space Filtering. Proc. 9th Int. Conf. Exp. Me-chanics, Copenhagen, 1990, pp. 1650–1659.

Subproject C2J. Hilbig, R. Ritter, W. Wilke: Hochtemperaturdehnungsmessung nach dem Rasterprinzip am

Beispiel des LCF-Versuchs. Akademie der Wissenschaften der DDR/Institut fur Mechanik, Re-port Nr. 24, 8, Chemnitz, 1989, pp. 255–258.

R. Ritter, W. Wilke: Optische in-situ-Dehnungsfeldmessung unter Hochtemperatureinfluss mit demRasterverfahren am Beispiel des LCF-Versuchs. Vortrags- und Diskussionstagung „Werkstoff-prufung 1990: Aussagefahigkeit von Prufungsergebnissen fur das Verarbeitungs- und Bauteil-verhalten“ 6./7. 12. 1990 Bad Nauheim, veranstaltet von DVM im Auftrag der Arbeitsge-meinschaft Werkstoffe.

J. Olfe, K.-T. Rie, R. Ritter, W. Wilke: In-situ-Messungen von Dehnungsfeldern bei Hochtempera-tur LCF. Zeitschrift fur Metallkunde 81(11) (1990) 783–789.

R. Ritter, J. Strusch, W. Wilke: Formanalyse mit Hilfe des Reflexions-Rasterverfahrens und desphotogrammetrischen Prinzips. Österreich. Ingenieur- und Architektenzeitung 135(7/8) (1990)346–348.

K. Galanulis, J. Hilbig, R. Ritter: Strain Measurement by the Diffraction Principle. Österreich. In-genieur- und Architektenzeitung 134(7/8) (1989) 392–394.

W. Cornelius, J. Hilbig, R. Ritter, W. Wilke, C. Forno: Zur Formanalyse mit Hilfe hochtempera-turbestandiger Raster. VDI-Bericht 731, 5, Dusseldorf, 1989, pp. 295–302.

K. Galanulis, J. Hilbig, B. W. Luhrig, R. Ritter: Strain Measurement by the Diffraction Principlefor Curved Surfaces. Proceedings of the 9th Int. Conference on Experimental Mechanics,Technical University of Denmark, Lyngby/Danemark, 20.–24. Aug. 1990.

Subproject C3P. Horst, H. Kossira: Zum Beulverhalten dunner Aluminiumplatten bei wechselnder Schubbelas-

tung. In: Proceedings der Jahrestagung der Deutschen Gesellschaft fur Luft- und Raumfahrt(DGLR), Jahrbuch 1988 I der DGLR, Bonn, 1988.

K. Wolf: FIPPS – Ein Programm-Paket zur numerischen Analyse des linearen und nichtlinearenTragverhaltens von Leichtbaustrukturen. In: H. Kossira (Ed.): 50 Jahre IFL, ZLR-Bericht 89-01, ISBN 3-9802073-0-7, Braunschweig, 1989.

K. Wolf: Untersuchungen zum Beul- und Nachbeulverhalten schubbeanspruchter Teilschalen auskohlenstoffaserverstarktem Kunststoff. Dissertation, Inst. f. Flugzeugbau und Leichtbau, Tech-nische Universitat Braunschweig, 1989.

P. Horst: Plastisches Beulen dunner Aluminiumplatten. In: H. Kossira (Ed.): 50 Jahre IFL, ZLR-Forschungsbericht 89-01, ISBN 3-9802073-0-7, Braunschweig, 1989.

P. Horst, H. Kossira: Theoretical and experimental investigation of thin-walled aluminium-panelsunder cyclic shearload. In: Proceedings of the International Conference on Spacecraft Struc-tures 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 the17th Congress of the International Council of the Aeronautical Sciences (ICAS) (Paper Nr. 90-4.4.1), Stockholm, 1990.

Bibliography

392

Subproject D1 (C4)J. Ruge, C.X. Hou, U. Patzold: Bestimmung von Gefugeinhomogenitaten in der Warmeeinfluss-

zone von Schweißverbindungen. Schweißen und Schneiden 41(3) (1989) 134–137.R. Linnemann: Beitrag zur Bewertung von Schweißnahtfehlern mittels bruchmechanischer Metho-

den. Fortschr.-Ber. VPI-Reihe 18, Nr. 55, VPI-Verlag, Dusseldorf, 1988.J. Ruge, S. Zhang, U. Patzold: Spannungsberechnungen in Schweißnahtmodellen mit Hilfe neuer

Werkstoffgesetze. Mat.-wiss. und Werkstofftech., Heft 9, 1990.

1991–1993

Subproject A1M. Zehetbauer, J. Schmidt, F. Haeßner: Calorimetric study of defect annihilation in low tempera-

ture 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 Microstruc-tures 13 (1991) 213.

J. Schmidt, F. Haeßner: Recovery and recrystallization of high purity lead determined within alow 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 andMaterials Society, 1992, pp. 233–257.

F. Haeßner, K. Sztwiertnia: Some microstructural aspects of the initial stage of recrystallization ofhighly rolled pure copper. Scripta Metallurgica et Materialia 27 (1992) 2933.

P. Kruger, E. Woldt: The use of an activation energy distribution for the analysis of the recrystal-lization kinetics of copper. Acta Metallurgica et Materialia 40 (1992) 2933.

E. Woldt: The relationship between isothermal and non-isothermal description of Johnson-Mehl-Avrami-Kolmogorov kinetics. J. Phys. Chem. Solids 53 (1992) 521.

F. Haeßner, J. Schmidt: Investigation of the recrystallization of low temperature deformed highlypure types of aluminium. Acta Metallurgica et Materialia 41 (1993) 1739.

K. Sztwiertnia, F. Haeßner: Orientation characteristics of the microstructure of highly pure cop-per and phosphorus copper. Textures and Microstructures 20 (1993) 87.

H.W. Hesselbarth, L. Kaps, F. Haeßner: Two dimensional simulation of the recrystallization kineticsin the case of inhomogeneously stored energy. Materials Science Forum 113–115 (1993) 317.

Subproject A2G. Lange, W. Gieseke: Veranderung des Werkstoffzustandes von Aluminium-Legierungen durch

mehrachsige plastische Wechselbeanspruchungen. Festschrift zur Vollendung des 65. Lebens-jahres 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 Instabili-tat. Z. f. Metallkunde 83 (1992) 115.

G. Lange: Schadensfalle durch Schwingbruche. Ingenieur-Werkstoffe 4(10) (1992) 62.G. Lange: Schwingbruche durch Steifigkeitssprunge. 15. Vortragsveranstaltung des Arbeitskreises

„Rastermikroskopie in der Materialprufung“, Deutscher Verband fur Materialforschung und-prufung, Berlin, 1992, pp. 285.

G. Lange: Schwingbruche durch konstruktiv bedingte Spannungsspitzen (Teil 1: SprunghafteQuerschnittsanderungen und Steifigkeitssprunge an Schweißverbindungen). Ingenieur-Werk-stoffe 5(3) (1993) 58.


393

G. Lange: Schwingbruche durch konstruktiv bedingte Spannungsspitzen (Teil 2: Verbindungs- undBefestigungselemente). Ingenieur-Werkstoffe 5(4) (1993) 74.

G. Lange: Fractures in Aircraft Components. In: H.P. Rossmanith, K. J. Miller (Eds.): Mixed-Mode Fatigue and Fracture, Mechanical Engineering Publications Limited, London, 1993, pp.23.

Subproject A5/B4K.-T. Rie, R. Schubert, H. Wittke: Cyclic Deformation Behaviour and Crack Growth in Low-Cy-

cle Fatigue Range. Mechanical Behaviour of Materials – VI, The Sixth International Confer-ence, Kyoto, Japan, Preprints of Additional Papers and Extended Abstracts, 1991, pp. 243–244.

K.-T. Rie, H. Wittke, R. Schubert: The �J-Integral and the Relation Between Deformation Behav-iour and Microstructure in the LCF-Range. In: K.-T. Rie (Ed.): Low Cycle Fatigue and Elas-tic-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, PergamonPress, 1991, pp. 367–372.

K.-T. Rie, J. Olfe: A Physically Based Model for Predicting LCF Life under Creep Fatigue Inter-action. In: K.-T. Rie (Ed.): Proc. 3rd Int. Conf. on Low Cycle Fatigue and Elastic-Plastic Be-haviour of Materials, Elsevier Applied Science, London/New York, 1992, pp. 222–228.

K.-T. Rie, J. Olfe: Dehnungsfelder vor Riss-Spitzen bei Kriechermudung. Z. Metallkde. 84(1993).

Subproject A6/B1E. Steck: Stochastic Models for the Plasticity of Metals. In: O. Bruller, 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 ShellStructures, 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 InelasticBehaviour. 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, Pro-ceedings of Plasticity ’91, Elsevier, London, 1991, pp. 175–178.

E. Steck: Stochastic Modelling of Cyclic Deformation Process in Metals (Reprint). In: S. I. Ander-sen 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 Experi-ments. 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 In-elastic Behaviour of Metals. Int. J. Plasticity 8 (1992) 147–169.

Subproject A8F. 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.

Bibliography

394

A. Kalk, Ch. Schwink: On sequences of alternate stable and unstable regions along tensile defor-mation 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. Neuhauser, 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 alongstress-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 crystalsoriented for single glide. phys. stat. sol. (a) 137 (1993) 337–350.

Subproject A9H. Neuhauser: Collective Dislocation Behaviour and Plastic Instabilities – Micro and Macro As-

pects. 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. Kruger, H. Neuhauser: Stress Relaxation in Cu-Al Single Crystals at High Tem-peratures. Scripta Metal. Mater. 27 (1992) 371–376.

J. Vergnol, F. Tranchant, A. Hampel, H. Neuhauser: Mesoscopic Observations Related to Twin-ning Instabilities in �-CuAl Crystals. In: O. Martin, L. Kubin (Eds.): Non-Linear Phenomenain Materials Science 11, Trans. Tech. Publ., Zurich, 1992, pp. 303–316.

H. Neuhauser, 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 De-formation and Fracture of Materials (Vol.-Ed.: H. Mughrabi), VCH Verlagsgemeinschaft,Weinheim, 1993, pp. 191–250.

A. Hampel, T. Kammler, H. Neuhauser: Structure and Kinetics of Luders Band Slip in Cu-5 to15at%Al Single Crystals. phys. stat. sol. (a) 135 (1993) 405–416.

H. Neuhauser: Collective Micro Shear Processes and Plastic Instabilities in Crystalline andAmorphous Structures. Int. J. Plasticity 9 (1993) 421–435.

C. Engelke, J. Plessing, H. Neuhauser: Plastic Deformation of Single Glide Oriented Cu-2 to15at%Al Crystals at Elevated Temperatures. Mater. Sci. Eng. A 164 (1993) 235–239.

H. Neuhauser: Problems in Solid Solution Hardening of Alloys. Physica Scripta T 49 (1993) 412–419.

H. Neuhauser, A. Hampel: Observation of Luders Bands in Single Crystals. Scripta Metal. Mater.29 (1993) 1151–1157.

Subproject A10D. Besdo: Eine Erweiterung der Taylor-Theorie zur Erfassung der kinematischen Verfestigung.

ZAMM-Z. Angew. Math. Mech. 71 (1991) T264–T265.D. Besdo, M. Muller: The Influence of Texture Development on the Plastic Behaviour of Polycrys-

tals. In: D. Besdo, E. Stein (Eds.): Finite Inelastic Deformations – Theory and Applications,IUTAM Symposium Hannover, Germany 1991, Springer Verlag, Berlin, Heidelberg, 1992.

M. Muller, D. Besdo: Simulation globaler Anisotropie mit Hilfe eines Vielkristallmodells. ZAMM-Z. Angew. Math. Mech. 73 (1993) T658.

Subproject B2G. Kracht: Erschließung viskoplastischer Stoffmodelle fur thermomechanische Strukturanalyse.

Report Nr. 93-69 from the Institut fur Statik der TU Braunschweig, 1993.E.-R. Tirpitz: Elasto-plastische Erweiterung von viskoplastischen Stoffmodellen fur Metalle. Re-

port Nr. 92-70 from the Institut fur Statik der TU Braunschweig, 1992.H. Braasch: Ein Konzept fur Fortentwicklung und Anwendung viskoplastischer Werkstoffmodelle.

Report Nr. 92-71 from the Institut fur Statik der TU Braunschweig, 1992.


395

L. Pisarsky: Zur Berechnung nichtmonoton beanspruchter wassergesattigter Tonboden. Report Nr.91-65 from the Institut fur Statik der TU Braunschweig, 1991.

Z. Huang: Beanspruchungen des Tunnelausbaus bei zeitabhangigem Materialverhalten von Betonund Gebirge. Report Nr. 91-68 from the Institut fur Statik der TU Braunschweig, 1991.

B. Hu: Berechnung des geometrisch und physikalisch nichtlinearen Verhaltens von Flachentrag-werken aus Stahl unter hohen Temperaturen. Report Nr. 93-72 from the Institut fur Statik derTU 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-In-dependent 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 Modellsfur reibungsbehafteten Kontakt. Report Nr. 93-75 from the Institut fur Statik der TU Braun-schweig, 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 Identifica-tion 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 zyk-lischer Versuche. Report Nr. 93-75 from the Institut fur Statik der TU Braunschweig, 1993,pp. 29–46.

H. Braasch: Erfassung streuenden Materialverhaltens in Werkstoffmodellen. Report Nr. 93-75from the Institut fur Statik der TU Braunschweig, 1993, pp. 1–14.

Subproject B5/B7J. Scheer, H.-J. Scheibe: Einachsige Zug-Druck-Versuche an Baustahl St52-3. Institut fur Stahl-

bau, TU Braunschweig, unpublished internal report.J. Scheer, H.-J. Scheibe: Untersuchungen von zyklisch beanspruchten Lochscheiben aus Baustahl

St52-3. Institut fur Stahlbau, TU Braunschweig, unpublished internal report.S. Dannemeyer: Verhalten von thermomechanisch behandelten Baustahlen unter zyklischer Bean-

spruchung im elastisch-plastischen Bereich. Experimentelle Studienarbeit, Institut fur Stahlbau,TU Braunschweig, 1992.

M.M. El-Ghandour: Low-Cycle Fatigue Damage Accumulation of Structural Steel St52. Disserta-tion TU Braunschweig, 1992.

L. Reifenstein: Verhalten modifizierter CT-Proben aus Baustahl St52-3 unter zyklischer Belastungim elastisch-plastischen Bereich. Experimentelle Studienarbeit, Institut fur Stahlbau, TU Braun-schweig.

U. Peil: Dynamisches Verhalten abgespannter Maste. VDI Bericht Nr. 924, 1992.

Subproject B6/B8E. Stein, G. Zhang, J.A. Konig: Shakedown with non-linear hardening including structural com-

putation 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 harden-ing 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 linearkinematic hardening materials. Int. J. of Solids and Structures 18 (1994) 2433–2444.

Bibliography

396

G. Zhang: Einspielen und dessen numerische Behandlung von Flachentragwerken aus ideal plas-tischem bzw. kinematisch verfestigendem Material. Ph. D. Thesis, Institut fur Baumechanikund Numerische Mechanik, Universitat Hannover, 1992.

R. Mahnken: Duale Verfahren fur nichtlineare Optimierungsprobleme in der Strukturmechanik.Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universitat Hannover,F92/3, 1992.

R. Mahnken, E. Stein: Parameter-Identification for Visco-Plastic Models via Finite-Element-Meth-ods 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 modelsin the context of finite elements methods. Proc. of Plasticity: Fourth Int. Symposium on Plastic-ity and its Current Applications, 1993.

Subproject C1K. Andresen, B. Hubner: 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 In-

ternational Conference on Photomechanics and Speckle Metrology, San Diego 1991, SPIE Pro-ceedings 1554 A (1991) 93–100.

K. Andresen, B. Kamp, R. Ritter: Three-dimensional surface deformation measurement by a grat-ing 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 directionmap. 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 fur Photogrammetrieund Fernerkundung 59 (1991) 212–220.

K. Andresen, R. Ritter, E. Steck: Theoretical and experimental investigations of crack extensionby FEM- and grating methods. Defect assessment in components. Fundamentals and applica-tion. ESIS/EGF9, Mechanical Engineering Publication, London, 1991, pp. 345–361.

Subproject C2R. Ritter, W. Wilke: Gliederung der Moireverfahren. Österreich. Ingenieur- und Architekten-Zei-

tung 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 Speckle-

Pattern 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, SPIEProceedings 1554A (1991) 588–592.

J.O. Hilbig, K. Galanulis, R. Ritter: Zur 3D-Verformungsmessung mit einem ElektronischenSpeckle-Interferometer. DVM-Tagungsband „Werkstoffprufung 1991“, Bad Nauheim, 1991, pp.103–110.

D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, D. Winter, H. Wohlfahrt: Zur Anwendung derSpeckle-Meßtechnik bei der Verformungsmessung in der Verbindungszone von Kaltpress-Schweißverbindungen verschiedener Werkstoffe. DVM-Tagungsband „Werkstoffprufung 1992“,Bad Nauheim, 1992, pp. 261–271.


397

D. Brinkmann, K. Galanulis, M. Kassner, Ritter, D. Winter, H. Wohlfahrt: Deformation analysisin the joining zone of cold pressure butt welding of different materials. International Sympo-sium on Mis-Matching of Welds, GKSS-Research Centre, Geesthacht, 1993.

D. Bergmann, B.-W. Luhrig, R. Ritter, D. Winter: Evaluation of ESPI-phase-images with regionaldiscontinuities. 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 discon-tinuities. International Workshop “Fringe”, Akademie Verlag, Berlin, 1993.

Subproject C3H. 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 dunner, bis in den plastischen Bereich zyklisch durch Schub belaste-

ter Aluminiumplatten. Dissertation, ZLR Forschungsbericht 91-01, ISBN 3-9802073-5-8, Inst.f. Flugzeugbau und Leichtbau, Technische Universitat Braunschweig, 1991.

P. Horst, H. Kossira, G. Arnst: On the Performance of Different Elastic-Plastic Material ModelsApplied to Cyclic Shear Buckling. Proc. of the Int. ECCS-Colloquium: On the Buckling ofShell 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 Com-bined Thermal and Mechanical Loads. Proc. of the Int. ECCS-Colloqium: On the Buckling ofShell 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 ofHypersonic 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 buck-ling behaviour of curved composite shear panels. Proceedings of the European Conferences onComposite Materials (ECCM), Amsterdam, 1992.

M. Haupt, H. Kossira: Integrated Thermal and Mechanical Structural Analysis of Hypersonic Ve-hicles by Using Adaptive Finite Element Methods. Proc. of the Third Aerospace Symposium1991, Braunschweig. In: Orbital Transport – Technical, Metereological and Chemical Aspects,Springer, 1993, pp. 165–178.

Subproject C4J. Ruge, U. Patzold: Einsatz einer vollautomatischen Harteprufstation zur Prufung von Schweiß-

verbindungen und Optimierung von Schweißverfahren. Tagungsband der DVM-Tagung Werk-stoffprufung, 1991.

U. Patzold: Verformungsanalyse von Schweißverbindungen. Dissertation, Institut fur Schweiß-technik, Technische Universitat Braunschweig, 1992.

D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, H. Wohlfahrt: Zur Anwendung der Speckle-Meßtechnik bei der Verformungsmessung in der Verbindungszone von Kaltpress-Schweißver-bindungen verschiedenartiger Werkstoffe. Tagungsband der DVM-Tagung Werkstoffprufung,1992, pp. 261–271.

D. Brinkmann, K. Galanulis, M. Kassner, R. Ritter, D. Winter, H. Wohlfahrt: Deformation Analy-sis in the Joining Zone of Cold Pressure Butt Welds of Different Materials. Proceedings of theConference on Mis-Matching of Welds, MEP, London, 1993.

Bibliography

398

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