Slip Systems edit page

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) → y↑→x)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  156.958 161.468 197.878
 
 
ans = slipSystem (y↑→x)
 
      x      y      z  |   x      y      z
  -0.51  -2.81  -0.93   0.03   0.06   -0.2