Plastic deformation in crystalline materials almost exclusively appears as dislocation along lattice planes. Such deformations are described by the normal vector n of the lattice plane and direction b of the slip. In the case of hexagonal alpha-Titanium with
cs = crystalSymmetry('622',[3,3,4.7],'x||a','mineral','Titanium (Alpha)')
cs = crystalSymmetry
mineral : Titanium (Alpha)
symmetry : 622
elements : 12
a, b, c : 3, 3, 4.7
reference frame: X||a, Y||b*, Z||c*
basal slip is defined by the Burgers vector (or slip direction)
b = Miller(2,-1,-1,0,cs,'UVTW')
b = Miller (Titanium (Alpha))
U V T W
2 -1 -1 0
and the slip plane normal
n = Miller(0,1,-1,0,cs,'HKIL')
n = Miller (Titanium (Alpha))
h k i l
0 1 -1 0
Putting both ingredients together we can define a slip system in MTEX by
sSBasal = slipSystem(b,n)
sSBasal = slipSystem (Titanium (Alpha))
U V T W | H K I L CRSS
2 -1 -1 0 0 1 -1 0 1
The most important slip systems for cubic, hexagonal and trigonal crystal lattices are already implemented into MTEX. Those can be accessed by
sSBasal = slipSystem.basal(cs)
sSBasal = slipSystem (Titanium (Alpha))
U V T W | H K I L CRSS
1 1 -2 0 0 0 0 1 1
Obviously, this is not the only basal slip system in hexagonal lattices. There are also symmetrically equivalent ones, which can be computed by
sSBasalSym = sSBasal.symmetrise('antipodal')
sSBasalSym = slipSystem (Titanium (Alpha))
size: 3 x 1
U V T W | H K I L CRSS
1 1 -2 0 0 0 0 1 1
1 -2 1 0 0 0 0 1 1
-2 1 1 0 0 0 0 1 1
The length of the burgers vector, i.e., the amount of displacement is
sSBasalSym.b.norm
ans =
3.0000
3.0000
3.0000
Displacement
In linear theory the displacement of a slip system is described by the strain tensor
sSBasal.deformationTensor
ans = tensor (Titanium (Alpha))
rank: 2 (3 x 3)
*10^-2
0 0 50
0 0 86.6
0 0 0
This displacement tensor is exactly the same as the so called Schmid tensor
sSBasal.SchmidTensor
ans = velocityGradientTensor (Titanium (Alpha))
rank: 2 (3 x 3)
*10^-2
0 0 50
0 0 86.6
0 0 0
Rotating slip systems
By definition the slip system and accordingly the deformation tensor are with the respect to the crystal coordinate system. In order to transform the quantities into specimen coordinates we have to multiply with some grain orientation
% some random grain orientation
ori = orientation.rand(cs)
% transfer slip system into specimen coordinates
ori * sSBasal
ori = orientation (Titanium (Alpha) → xyz)
Bunge Euler angles in degree
phi1 Phi phi2
156.958 161.468 197.878
ans = slipSystem (xyz)
x y z | x y z
-0.51 -2.81 -0.93 0.03 0.06 -0.2