Microstructure Evolution in Crystal Plasticity

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    Microstructure evolution in crystal plasticity: strain path effects and dislocation slip

    patterning

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    This research was carried out under the project number MC2.03158 in the framework of 

    the Research Program of the Materials innovation institute M2i (www.m2i.nl), the former

    Netherlands Institute for Metals Research.

    CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

    Tuncay Yalçınkaya

    Microstructure evolution in crystal plasticity: strain path effects and dislocation slip patterning /

     by T. Yalçınkaya – Eindhoven : Technische Universiteit Eindhoven, 2011. Proefschrift.

    A catalogue record is available from the Eindhoven University of Technology Library ISBN: 978-90-386-2729-8 Subject headings: BCC metals / crystal plasticity / non-Schmid effects / plastic anisotropy / strain path change effect / Bauschinger effect / cross effect / microstructure evolution / non-convexity / phase field modeling / dislocation patterning / finite element method / non-convex free energy / strain gradient crystal plasticity Copyright   c2011 by Tuncay Yalçınkaya, all rights reserved.

    This thesis was prepared with the LATEX 2ε documentation system.

    Reproduction: Universiteitsdrukkerij TU Eindhoven, Eindhoven, The Netherlands.

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    Microstructure evolution in crystal plasticity: strain path effects and dislocation slip

    patterning

    PROEFSCHRIFT

    ter verkrijging van de graad van doctor

    aan de Technische Universiteit Eindhoven,

    op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn,

    voor een commissie aangewezen door het College voor Promoties

    in het openbaar te verdedigen

    op donderdag 20 oktober 2011 om 16.00 uur

    door

    Tuncay Yalçınkaya

    geboren te Ankara, Turkije

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    Dit proefschrift is goedgekeurd door de promotor:

    prof.dr.ir. M.G.D. Geers

    Copromotor:

    dr.ir. W.A.M. Brekelmans

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    Contents

    Summary ix

    1 Introduction 1

    1.1 Crystal plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 Objective and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    2 A finite strain BCC single crystal plasticity model and its experimental

    identification 5

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2.2 Slip mechanisms in BCC metals . . . . . . . . . . . . . . . . . . . . . . . 8

    2.3 Violation of Schmid’s law in BCC metals . . . . . . . . . . . . . . . . . . 10 2.4 A BCC crystal plasticity model at material point level . . . . . . . . . . 12

    2.4.1 Kinematics in crystal plasticity . . . . . . . . . . . . . . . . . . . 12

    2.4.2 Constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2.5 Modeling some intrinsic properties of BCC single crystals . . . . . . . . 17

    2.5.1 Orientation dependence . . . . . . . . . . . . . . . . . . . . . . . 17

    2.5.2 Example: α -Fe single crystal . . . . . . . . . . . . . . . . . . . . 18

    2.5.3 Example: molybdenum single crystal . . . . . . . . . . . . . . . 19

    2.5.4 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . 19

    2.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    3 A composite dislocation cell model to describe strain path change effects in

    BCC metals 25

    3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    3.2 Dislocation substructure evolution . . . . . . . . . . . . . . . . . . . . . 28

    3.3 Computational model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    3.4 Modeling of microstructure evolution . . . . . . . . . . . . . . . . . . . 34

    3.4.1 Monotonic deformation . . . . . . . . . . . . . . . . . . . . . . . 34

    3.4.2 Orthogonal change of deformation . . . . . . . . . . . . . . . . . 36

    3.4.3 Reverse deformation . . . . . . . . . . . . . . . . . . . . . . . . . 37

    v

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    vi   Contents

    3.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    3.5.1 Example 1: monotonic deformation of single crystals . . . . . . 38

    3.5.2 Example 2: strain path change of single crystals . . . . . . . . . 39 3.5.3 Example 3: strain path change of polycrystals . . . . . . . . . . 40

    3.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    4 Deformation patterning driven by rate dependent non-convex strain gradi-

    ent plasticity 45

    4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    4.2 Macroscopic view: material instability and microstructure evolution

    in inelastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    4.3 Thermodynamics of strain gradient plasticity . . . . . . . . . . . . . . . 50

    4.4 Particular choices of free energy functions . . . . . . . . . . . . . . . . . 54

    4.4.1 Slip based strain gradient plasticity . . . . . . . . . . . . . . . . 54

    4.4.2 Slip based non-convex strain gradient plasticity . . . . . . . . . 55

    4.5 Non-convexity and patterning in phase field modeling . . . . . . . . . 57

    4.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    4.6.1 Numerical example 1: convex case - monotonic loading . . . . . 59

    4.6.2 Numerical example 2: non-convex case - monotonic loading . . 60

    4.6.3 Numerical example 3: non-convex stress relaxation of a 1D bar 66

    4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    4.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.8.1 Finite element implementation of slip based strain gradient

    plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    4.8.2 Finite element implementation of slip based non-convex strain

    gradient plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 71

    5 Non-convex rate dependent strain gradient crystal plasticity and deforma-

    tion patterning 73

    5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    5.2 Strain gradient crystal plasticity and finite element implementation . . 755.3 Latent hardening based non-convex plastic potential . . . . . . . . . . . 80

    5.3.1 Conditions for plastic slip patterning . . . . . . . . . . . . . . . 80

    5.4 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    5.4.1 Convex strain gradient crystal plasticity . . . . . . . . . . . . . . 84

    5.4.2 Non-convex strain gradient crystal plasticity . . . . . . . . . . . 88

    5.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    6 Discussion and conclusions 95

    Bibliography 99

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    Contents   vii

    Dankwoord / Acknowledgements 109

    Curriculum Vitae 111

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    viii

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    Summary

    During deformation polycrystalline metals tend to develop heterogeneous plastic deformation fields at the microscopic scale, as the amount of plastic strain varies

    spatially, depending on local grain orientation, geometry and defects. While grain

     boundaries are natural places triggering plastic slip accumulation and geometrically

    necessary dislocations that accommodate the gradients of the inhomogeneous plastic

    strain, the deformation localizes within grains revealing dislocation cell structures or

    micro slip bands (e.g. clear band formation in irradiated materials). Across grains,

    macroscopic plastic slip bands (Lüders bands, etc.) exist as well. These intergran-

    ular and intragranular deformation patterns are stated to be inherent minimizers

    of the free energy (including the microstructurally trapped plastic energy). These microstructures may macros