The Vertex effect in polycrystalline materials [Elektronische Ressource] : simulation, a macroscopic model, and structural application / von Michael Schurig

Informations

• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2006
• Nombre de lectures: 35
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

The Vertex Effect in Polycrystalline Materials
Simulation, a Macroscopic Model, and Structural Application
Dissertation
zur Erlangung des akademischen Grades
Doktoringenieur
(Dr.-Ing.)
von Dipl.-Ing. Michael Schurig
geb. am 16.01.1973 in Berlin
genehmigt durch die Fakulta¨t fur¨ Maschinenbau
der Otto-von-Guericke-Universita¨t Magdeburg
Gutachter:
Prof. Albrecht Bertram
Prof. Henryk Petryk
Promotionskolloquium am 17.07.2006The Vertex Effect in Polycrystalline Materials
Simulation, a Macroscopic Model, and Structural ApplicationThe Vertex Effect in Polycrystalline Materials
Simulation, a Macroscopic Model, and Structural Application
Dissertation
zur Erlangung des akademischen Grades
Doktoringenieur
(Dr.-Ing.)
von Dipl.-Ing. Michael Schurig
geb. am 16.01.1973 in Berlin
genehmigt durch die Fakulta¨t fur¨ Maschinenbau
der Otto-von-Guericke-Universita¨t Magdeburg
Gutachter:
Prof. Albrecht Bertram
Prof. Henryk Petryk
Promotionskolloquium am 17.07.2006
September 5, 20068
Acknowledgements
Soli Deo Gloria.
Theworkpresentedinthisthesishasbeendoneduringmytimeasaresearch
staff member at the Otto-von-Guericke-Universit¨at Magdeburg.
I am indebted to Prof. Albrecht Bertram, who was the advisor and the
first referee of this thesis. He was a guide in finding the topic of the work and
finally finishing it. His mentorship by teaching the theoretical foundations of
continuummechanicsandwideningtheviewbystimulatingavarietyofcontacts
was a great experience.
I also wish to thank Prof. Henryk Petryk, Polish Academy of Science, for
the supportive discussions on the general topic and several aspects of the work,
and for being the second referee.
Ienjoyedbeingamemberofateamofgreatcolleagues. Inparticular, Iwish
to thank Enrico Brosche for his help with numerical integration.
I would like to mention the support and the interdisciplinary discussions
as an associated member in the Graduate School ”Micro-Macro-Interactions in
Structured Media and Particle Systems”, supported by DFG.
Last but not least I wish to express my deepest gratitude to my beloved
wife Doris, who cared for our children Lisa and Julius and by this gave me the
opportunity to work on this thesis.CONTENTS
1. Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 The Vertex Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Isotropic criteria . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Early models of polycrystal plasticity . . . . . . . . . . . 15
2.2.3 Experimental research . . . . . . . . . . . . . . . . . . . . 16
2.2.4 Computational approaches . . . . . . . . . . . . . . . . . 17
2.2.5 Construction of material models . . . . . . . . . . . . . . 18
2.2.6 Application . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Overview and main results . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Part I Theoretical Foundations 23
3. General aspects of continuum mechanics . . . . . . . . . . . . . . . . . 25
3.1 Kinematics of continua . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Placement and motion . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Deformation and strain measures . . . . . . . . . . . . . . 26
3.1.3 Change of reference placement . . . . . . . . . . . . . . . 27
3.2 Dynamics of continua . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Spatial balance equations . . . . . . . . . . . . . . . . . . 28
3.2.2 Material balance equations . . . . . . . . . . . . . . . . . 28
3.2.3 Work conjugate stress and strain measures . . . . . . . . 29
3.3 The second law of thermodynamics . . . . . . . . . . . . . . . . . 30
4. Rate independend plastic materials . . . . . . . . . . . . . . . . . . . . 31
4.1 Elastic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.1 Hyperelastic laws . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.2 Principle of material objectivity. . . . . . . . . . . . . . . 31
4.1.3 Change of strain measure and reference placement . . . . 32
4.1.4 The general hyperelastic law . . . . . . . . . . . . . . . . 32
4.1.5 Linear elastic laws . . . . . . . . . . . . . . . . . . . . . . 32
4.1.6 Rate forms of the elastic law . . . . . . . . . . . . . . . . 3410 Contents
4.1.7 Elastic isomorphy . . . . . . . . . . . . . . . . . . . . . . 35
4.1.8 Elastic symmetry . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Theory of plasticity based on isomorphisms . . . . . . . . . . . . 36
4.2.1 Isomorphisms between elastic laws . . . . . . . . . . . . . 36
4.2.2 The multiplicative split . . . . . . . . . . . . . . . . . . . 37
4.2.3 Kinematic split of velocity gradient . . . . . . . . . . . . . 37
4.2.4 Small elastic deformations . . . . . . . . . . . . . . . . . . 39
4.3 Constitutive theory of multi-mode plasticity . . . . . . . . . . . . 39
4.3.1 Multiple mechanisms of plasticity . . . . . . . . . . . . . . 39
4.3.2 Solution of the DAE . . . . . . . . . . . . . . . . . . . . . 41
4.3.3 Properties of the multiple condition of consistency . . . . 44
4.3.4 The loading condition . . . . . . . . . . . . . . . . . . . . 45
4.3.5 Derivation of rate equations . . . . . . . . . . . . . . . . . 46
4.3.6 Rate-independence . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Regular yield criteria . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.1 The model of interacting slip systems . . . . . . . . . . . 48
4.4.2 The viscoplastic flow potential . . . . . . . . . . . . . . . 52
4.4.3 Regular yield surfaces for multiple mechanisms . . . . . . 53
4.5 The vertex effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5.1 Straight and kinked loading . . . . . . . . . . . . . . . . . 54
4.5.2 Symmetric planar double slip . . . . . . . . . . . . . . . . 56
4.5.3 Regularized plastic potential and multislip . . . . . . . . . 61
4.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Part II Plasticity with yield vertices 65
5. Simulation-based analysis of yield-vertex development . . . . . . . . . 67
5.1 Micromechanical Investigation of subsequent yield surfaces. . . . 67
5.1.1 Usage of the Taylor-Lin model . . . . . . . . . . . . . . . 68
5.1.2 The viscous flow potential as a yield indicator . . . . . . . 70
5.1.3 The dissipated power as a path dependend yield criterion 70
5.1.4 The initial yield surface . . . . . . . . . . . . . . . . . . . 70
5.1.5 Subsequent yield surfaces . . . . . . . . . . . . . . . . . . 72
5.2 Development of the outer corner angle . . . . . . . . . . . . . . . 81
5.3 Complex strain processes . . . . . . . . . . . . . . . . . . . . . . 84
5.3.1 Shear after unloaded tension . . . . . . . . . . . . . . . . 84
5.3.2 Shear immediately after tension . . . . . . . . . . . . . . . 84
5.4 Consequences for a macroscopic vertex model . . . . . . . . . . . 84
6. A macroscopic vertex model . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Process dependence of dissipation . . . . . . . . . . . . . . . . . . 89
6.2 Process dep of the flow rule . . . . . . . . . . . . . . . . . 91
6.2.1 The associated elastic domain . . . . . . . . . . . . . . . . 92
6.2.2 The hardening rule . . . . . . . . . . . . . . . . . . . . . . 93
6.2.3 Exploitation of the condition of consistency . . . . . . . . 93