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-Hill
Many 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