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Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
Lecture Micromechanics texture SFB 761
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Lecture Micromechanics texture SFB 761

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  • 1. Why does a crystal rotate ? Dierk Raabe Düsseldorf, Germany WWW.MPIE.DE d.raabe@mpie.de SFB Class 2012
  • 2. Overview Roters et al. Acta Materi.58 (2010) 1
  • 3. 2
  • 4. Some dislocation kinematics and kinetics: phenomena true strain stress 3
  • 5. Plastic deformation of a single crystal by dislocation slip 4
  • 6. Plastic deformation of a single crystal by dislocation slip 5
  • 7. Plastic deformation of a single crystal by dislocation slip 6
  • 8. Plastic deformation of a single crystal by dislocation slip d dx b 1   n   m bv dt X Z dt 7
  • 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. Plastic deformation of a single crystal by dislocation slip
  • 11. Plastic deformation of a single crystal by dislocation slip
  • 12. Boundary condition: determines lab frame constraints 11
  • 13. Single crystal plasticity: crystal shear and crystal rotation 12
  • 14. Plastic deformation of a single crystal by dislocation slip d dx b 1   n   m bv dt X Z dt 13
  • 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. 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. Kinematics, displacement, displacement gradient: general 16
  • 18. Kinematics: Micro-to-macro-transition 17
  • 19. Geometrical interpretation ? 18
  • 20. Geometrical interpretation 19
  • 21. Complex boundary conditions mesoscopic boundary conditions one dislocation (grain / orientation neighborhood) parallel loops reactions orientation change 20
  • 22. Simplify boundary conditions Boundary conditions: 1) Upper bound treatment: iso-stress 2) Lower bound treatment: iso-strain iso strain 21
  • 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. Single crystal plasticity: multiple slip (or twinning) system /mit /mit 23
  • 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. 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. 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. Simplify boundary conditions Boundary conditions: 1) Upper bound treatment: iso-stress 2) Lower bound treatment: iso-strain iso strain 27
  • 29. The Taylor Model 28
  • 30. Crystal yield surface, Taylor Bishop-Hill 29
  • 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. Homogeneity and boundary conditions – meso-scale 3% 8% 15%
  • 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. Multiscale crystal plasticity FEM or FFT Raabe, Zhao, Park, Roters: Acta Mater. 50 (2002) 421 33
  • 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

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