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# Lecture Micromechanics texture SFB 761

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### Lecture Micromechanics texture SFB 761

1. 1. Why does a crystal rotate ? Dierk Raabe Düsseldorf, Germany WWW.MPIE.DE d.raabe@mpie.de SFB Class 2012
2. 2. Overview Roters et al. Acta Materi.58 (2010) 1
3. 3. 2
4. 4. Some dislocation kinematics and kinetics: phenomena true strain stress 3
5. 5. Plastic deformation of a single crystal by dislocation slip 4
6. 6. Plastic deformation of a single crystal by dislocation slip 5
7. 7. Plastic deformation of a single crystal by dislocation slip 6
8. 8. Plastic deformation of a single crystal by dislocation slip d dx b 1   n   m bv dt X Z dt 7
9. 9. Plastic deformation of a single crystal by dislocation slip Schmid factor S h id f t (orientation factor for that slip system) 8
10. 10. Plastic deformation of a single crystal by dislocation slip
11. 11. Plastic deformation of a single crystal by dislocation slip
12. 12. Boundary condition: determines lab frame constraints 11
13. 13. Single crystal plasticity: crystal shear and crystal rotation 12
14. 14. Plastic deformation of a single crystal by dislocation slip d dx b 1   n   m bv dt X Z dt 13
15. 15. Kinematics, displacement u u(x,y,z) u=u(x y z) (x(1),y,z) (x(2),y,z) 1 2 u(1)(x,y,z) u(2)(x,y,z)u(1)(x y z)=u(2)(x y z) (x,y,z)=u (x,y,z) 1 2 1 2 14
16. 16. Kinematics, displacement u u(x,y,z) u=u(x y z) (x(1),y,z) (x(2),y,z) 1 2 u(1)(x,y,z) u(2)(x,y,z)u(1)(x y z)≠u(2)(x y z) (x,y,z)≠u (x,y,z) 1 2 1 2 15
17. 17. Kinematics, displacement, displacement gradient: general 16
18. 18. Kinematics: Micro-to-macro-transition 17
19. 19. Geometrical interpretation ? 18
20. 20. Geometrical interpretation 19
21. 21. Complex boundary conditions mesoscopic boundary conditions one dislocation (grain / orientation neighborhood) parallel loops reactions orientation change 20
22. 22. Simplify boundary conditions Boundary conditions: 1) Upper bound treatment: iso-stress 2) Lower bound treatment: iso-strain iso strain 21
23. 23. Iso-stress: single slip system  a ma sym  b mbsym  d md sym  c mcsym    1  c  b   d   a  krit  krit  krit  krit D  33 T T D a D b c D  krit aktiv d  11 22
24. 24. Single crystal plasticity: multiple slip (or twinning) system /mit /mit 23
25. 25. Single crystal yield surface 33 1 crystal, 1 slip system: aik nk a jlbl  ijj   crit j slip system 1 ..  crit s 1 ( active) same strain 33 different 11 stresses slip system 2  crit s  2 ( active)  crit active 1 crystal, 2 slip systems: aik nks a jlbls ij   crit  11
26. 26. Iso-stress: multiple slip (or twinning) system 33 Ds=1 Ds=2  Vers ..  krit (+) s1 ( aktiv) krit,(+)  TBH S 11  krit,(-) s2  krit,(+) s2 ( aktiv) .  krit,(-) s1 ( aktiv)  Vers Ds=2
27. 27. Single crystal plasticity bcc, fcc, bcc fcc Bcc: 24 systems Section in stress spacekrit krit  krit krit BCC, 48 systemskrit krit 26
28. 28. Simplify boundary conditions Boundary conditions: 1) Upper bound treatment: iso-stress 2) Lower bound treatment: iso-strain iso strain 27
29. 29. The Taylor Model 28
30. 30. Crystal yield surface, Taylor Bishop-Hill 29
31. 31. Crystal yield surface, Taylor Bishop-HillMany crystals, many slip systems: crystals grain 1 33 grain 2 g grain 3 grain 4 imposed strain 11
32. 32. Homogeneity and boundary conditions – meso-scale 3% 8% 15%
33. 33. Simplify boundary conditions Boundary conditions: 1) Upper bound treatment: iso-stress (strain not compatible) 2) Lower bound treatment: iso-strain (forces not in equilibrium) 32
34. 34. Multiscale crystal plasticity FEM or FFT Raabe, Zhao, Park, Roters: Acta Mater. 50 (2002) 421 33
35. 35. Crystal Mechanics FEM, grain scale mechanics (2D) Experiment (DIC, EBSD) v Mises strain Simulation (CP-FEM) (C ) v Mises strain Sachtleber, Sachtleber, Zhao, Raabe: Mater. Sc. Engin. A 336 (2002) 81 Mater. Engin. 34