More Related Content More from Dierk Raabe (20) Lecture Micromechanics texture SFB 7611. 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
9. Plastic deformation of a single crystal by dislocation slip
Schmid factor
S h id f t
(orientation factor for that slip system)
8
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
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
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 (+)
s1 ( aktiv)
krit,(+)
TBH
S
11
krit,(-)
s2
krit,(+)
s2 ( aktiv)
.
krit,(-)
s1 ( 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
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
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
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