Slip Systems

Crystal slip shears one part of a crystal past another by dislocation motion on a lattice plane. A slip system specifies the plane and the direction of that shear. This page constructs one system, generates its symmetry-equivalent family, and assigns strengths to several families.

Define one slip system

A slip system combines a Burgers vector \(\mathbf b\), which gives the slip direction and displacement, with a slip-plane normal \(\mathbf n\). The direction must lie in the plane, so \(\mathbf b\cdot\mathbf n=0\).

Start with the lattice and crystal frame of hexagonal alpha-titanium. The four-index notation used below is introduced with crystal directions.

cs = crystalSymmetry('622',[3,3,4.7],'x||a', ...
  'mineral','Titanium (Alpha)')

cs = crystalSymmetry (⊙c→a)
 
  mineral        : Titanium (Alpha) 
  symmetry       : 622              
  elements       : 12               
  a, b, c        : 3, 3, 4.7        
  reference frame: X||a, Y||b*, Z||c

One first-order prismatic \(\langle a\rangle\) system has Burgers vector \([2\bar1\bar10]\) and plane normal \((01\bar10)\).

b = Miller(2,-1,-1,0,cs,'UVTW')
n = Miller(0,1,-1,0,cs,'HKIL')

b = Miller (Titanium (Alpha))
  U  V  T  W
  2 -1 -1  0
 
n = Miller (Titanium (Alpha))
  h  k  i  l
  0  1 -1  0

Passing the two directions to the slipSystem constructor keeps them together as one physical shear mode. The constructor also checks orthogonality.

sSPrismatic = slipSystem(b,n)

sSPrismatic = slipSystem (Titanium (Alpha))
 
  U    V    T    W  | H    K    I    L CRSS
  2   -1   -1    0    0    1   -1    0    1

Common families also have named constructors. For example, this creates one representative of the basal \(\langle11\bar20\rangle\{0001\}\) family.

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

Draw the plane and direction

Drawn inside the crystal, the plane shows where the lattice shears and the arrow shows the direction of shear. The same plane with another in-plane direction is therefore a different slip system.

cS = crystalShape.hex(cs);

plot(cS,'faceAlpha',0.4,'faceColor',[0.7 0.8 0.9])
hold on
plot(cS,sSBasal,'faceColor','red')
hold off

The red disk is the basal plane. The red arrow lies in that disk, which makes the required orthogonality of \(\mathbf b\) and \(\mathbf n\) visible.

Generate the complete family

A representative does not describe every basal system in a hexagonal crystal. Crystal symmetry generates the equivalent systems. The option 'antipodal' identifies opposite Burgers-vector signs, because they describe the two shear senses of the same geometric system.

sSBasalSym = sSBasal.symmetrise('antipodal')

sSBasalSym = slipSystem (Titanium (Alpha))
 size: 3 × 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

Alpha-titanium has three such basal systems. The norm of each Burgers vector is 3 in the lattice units selected in cs.

length(sSBasalSym)
sSBasalSym.b.norm

ans =
     3
ans =
     3
     3
     3

Choose families and their CRSS

For cubic lattices, slipSystem.fcc(cs) and slipSystem.bcc(cs) provide standard sets. Hexagonal lattices deliberately have no slipSystem.hcp. The active families and their critical resolved shear stress (CRSS) depend on the material, temperature, and loading rather than on the lattice alone. MTEX therefore provides each family separately:

slipSystem.basal(cs)          <11-20>{0001}
slipSystem.prismaticA(cs)     <2-1-10>{01-10}
slipSystem.prismatic2A(cs)    <01-10>{2-1-10}     2nd order prismatic
slipSystem.pyramidalA(cs)     <2-1-10>{01-11}     1st order pyramidal <a>
slipSystem.pyramidalCA(cs)    <2-1-13>{-1101}     1st order pyramidal <c+a>
slipSystem.pyramidal2CA(cs)   <2-1-13>{-2112}     2nd order pyramidal <c+a>
slipSystem.twinT1(cs)         <1-101>{-1102}      tensile twinning
slipSystem.twinT2(cs)         <2-1-16>{-2111}     tensile twinning
slipSystem.twinC1(cs)         <-110-2>{-1101}     compressive twinning
slipSystem.twinC2(cs)         <2-1-1-3>{2-1-12}   compressive twinning

The second argument sets the CRSS of a family. This illustrative set makes the basal systems easiest to activate and makes the families comparable. Use values measured for the material and conditions in a real model.

sS = [slipSystem.basal(cs,1), slipSystem.prismatic2A(cs,66), ...
  slipSystem.pyramidalCA(cs,80), slipSystem.twinC1(cs,100)]

sS = slipSystem (Titanium (Alpha))
 size: 1 × 4
 
   U    V    T    W  | H    K    I    L CRSS
   1    1   -2    0    0    0    0    1    1
   0    1   -1    0    2   -1   -1    0   66
   2   -1   -1    3   -1    1    0    1   80
  -1    1    0   -2   -1    1    0    1  100

Deformation and Schmid tensors

In linearized kinematics, a unit shear on a slip system contributes the displacement gradient \(\mathbf b\otimes\mathbf n\) after both vectors are normalized. Its symmetric part is strain and its antisymmetric part is lattice spin. MTEX returns this quantity as the deformation tensor.

L = sSBasal.deformationTensor

L = tensor (Titanium (Alpha))
  rank: 2 (3 × 3)
 
 *10^-2
    0    0   50
    0    0 86.6
    0    0    0

MTEX uses exactly the same normalized dyad as the Schmid tensor. The next page contracts it with a stress tensor to obtain resolved shear stress.

S = sSBasal.SchmidTensor

S = velocityGradientTensor (Titanium (Alpha))
  rank: 2 (3 × 3)
 
 *10^-2
    0    0   50
    0    0 86.6
    0    0    0

Express a system in the specimen frame

A newly constructed slip system is expressed in the crystal frame. An orientation maps the crystal frame into a specimen frame. Multiplication applies that map to both \(\mathbf b\) and \(\mathbf n\).

ori = orientation.rand(cs)
sSSpecimen = ori * sSBasal

ori = orientation (Titanium (Alpha) → y↓→x)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  156.958 161.468 197.878
 
 
sSSpecimen = slipSystem (y↓→x)
 
      x      y      z  |   x      y      z
  -0.51  -2.81  -0.93   0.03   0.06   -0.2

References

Next

Continue with Schmid Factor to relate the plane and direction of each slip system to an applied stress.