Schmid Factor Analysis

This script describes how to analyze Schmid factors.

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Geometric Definition
Stress Tensor
Active Slip System
The Schmid factor for EBSD data

Geometric Definition

Let us assume a Nickel crystal

CS = crystalSymmetry('cubic',[3.523,3.523,3.523],'mineral','Nickel')
 
CS = crystalSymmetry  
 
  mineral : Nickel       
  symmetry: m-3m         
  a, b, c : 3.5, 3.5, 3.5
 

Since Nickel is fcc a dominat slip system is given by the slip plane normal

n = Miller(1,1,1,CS,'hkl')
 
n = Miller  
 size: 1 x 1
 mineral: Nickel (m-3m)
  h 1
  k 1
  l 1

and the slip direction (which needs to be orthogonal)

d = Miller(0,-1,1,CS,'uvw')
 
d = Miller  
 size: 1 x 1
 mineral: Nickel (m-3m)
  u  0
  v -1
  w  1

For a simple shear in the z - direction

r = normalize(vector3d(1,2,3))
 
r = vector3d  
 size: 1 x 1
         x        y        z
  0.267261 0.534522 0.801784

the Schmid factor for the slip system [0-11](111) is defined by

tau = dot(d,r,'noSymmetry') * dot(n,r,'noSymmetry')
tau =
    0.4286

The same computation can be performed by defining the slip system as an MTEX variable

sS = slipSystem(d,n)
 
sS = slipSystem  
 mineral: Nickel (m-3m)
 CRSS: 1
 size: 1 x 1
  u   v   w | h   k   l
  0  -1   1   1   1   1

and using the command SchmidFactor

sS.SchmidFactor(r)
ans =
    0.1750

Ommiting the tension direction r the command SchmidFactor returns the Schmid factor as a spherical function

%will be part of MTEX 4.5
% SF = sS.SchmidFactor

% plot the Schmid factor in dependency of the tension direction
%will be part of MTEX 4.5
%plot(SF)

%will be part of MTEX 4.5
%[SFMax,pos] = max(SF)

Stress Tensor

Instead by the tension direction the stress might be specified by a stress tensor

sigma = tensor([0 0 0; 0 0 0; 0 0 1],'name','stress')
 
sigma = stress tensor  
  rank: 2 (3 x 3)
 
 0 0 0
 0 0 0
 0 0 1

Then the Schmid factor for the slip system sS and the stress tensor sigma is computed by

sS.SchmidFactor(sigma)
ans =
    0.4082

Active Slip System

In general a crystal contains not only one slip system but at least all symmetrically equivalent ones. Those can be computed with

sSAll = sS.symmetrise('antipodal')
 
sSAll = slipSystem  
 mineral: Nickel (m-3m)
 CRSS: 1
 size: 12 x 1
   u   v   w | h   k   l
   0  -1   1   1   1   1
   1   0  -1   1   1   1
  -1   1   0   1   1   1
  -1   1   0   1   1  -1
  -1   0  -1   1   1  -1
   0  -1  -1   1   1  -1
   0  -1   1  -1   1   1
  -1   0  -1  -1   1   1
  -1  -1   0  -1   1   1
   1   0  -1   1  -1   1
  -1  -1   0   1  -1   1
   0  -1  -1   1  -1   1

The option antipodal indicates that Burgers vectors in oposite direction should not be distinguished. Now

tau = sSAll.SchmidFactor(r)
tau =
  Columns 1 through 7
    0.1750   -0.3499    0.1750    0.0000   -0.0000   -0.0000    0.1166
  Columns 8 through 12
   -0.4666   -0.3499   -0.1166   -0.1750   -0.2916

returns a list of Schmid factors and we can find the slip system with the largest Schmid factor using

[tauMax,id] = max(abs(tau))

sSAll(id)
tauMax =
    0.4666
id =
     8
 
ans = slipSystem  
 mineral: Nickel (m-3m)
 CRSS: 1
 size: 1 x 1
   u   v   w | h   k   l
  -1   0  -1  -1   1   1

The above computation can be easily extended to a list of tension directions

% define a grid of tension directions
r = plotS2Grid('resolution',0.5*degree,'upper')

% compute the Schmid factors for all slip systems and all tension
% directions
tau = sSAll.SchmidFactor(r);

% tau is a matrix with columns representing the Schmid factors for the
% different slip systems. Lets take the maximum rhowise
[tauMax,id] = max(abs(tau),[],2);

% vizualize the maximum Schmid factor
contourf(r,tauMax)
mtexColorbar
 
r = vector3d  
 size: 181 x 721
 resolution: 0.5°
 plot: true              
 region: upper hemisphere
 theta: 181 x 721 double 
 rho: 181 x 721 double   
e = 
  PropertyEvent with properties:

    AffectedObject: [1×1 ColorBar]
            Source: [1×1 matlab.graphics.internal.GraphicsMetaProperty]
         EventName: 'PostSet'

We may also plot the index of the active slip system

pcolor(r,id)

mtexColorMap black2white

and observe that within the fundamental sectors the active slip system remains the same. We can even visualize the the plane normal and the slip direction

% if we ommit the option antipodal we can distinguish
% between the oposite burger vectors
sSAll = sS.symmetrise

% take as directions the centers of the fundamental regions
r = symmetrise(CS.fundamentalSector.center,CS);

% compute the Schmid factor
tau = sSAll.SchmidFactor(r);

% here we do not need to take the absolut value since we consider both
% burger vectors +/- b
[~,id] = max(tau,[],2);

% plot active slip plane in red
hold on
quiver(r,sSAll(id).n,'ArrowSize',0.2,'LineWidth',2,'Color','r');

% plot active slip direction in green
hold on
quiver(r,sSAll(id).b.normalize,'ArrowSize',0.1,'LineWidth',2,'Color','g');
hold off
 
sSAll = slipSystem  
 mineral: Nickel (m-3m)
 CRSS: 1
 size: 24 x 1
   u   v   w | h   k   l
   0  -1   1   1   1   1
   1   0  -1   1   1   1
  -1   1   0   1   1   1
   0   1  -1   1   1   1
  -1   0   1   1   1   1
   1  -1   0   1   1   1
  -1   1   0   1   1  -1
  -1   0  -1   1   1  -1
   0  -1  -1   1   1  -1
   1   0   1   1   1  -1
   0   1   1   1   1  -1
   1  -1   0   1   1  -1
   0  -1   1  -1   1   1
  -1   0  -1  -1   1   1
  -1  -1   0  -1   1   1
   1   0   1  -1   1   1
   1   1   0  -1   1   1
   0   1  -1  -1   1   1
   1   0  -1   1  -1   1
  -1  -1   0   1  -1   1
   0  -1  -1   1  -1   1
   1   1   0   1  -1   1
   0   1   1   1  -1   1
  -1   0   1   1  -1   1

The Schmid factor for EBSD data

So far we have always assumed that the stress tensor is already given relatively to the crystal coordinate system. Next, we want to examine the case where the stress is given in specimen coordinates and we know the orientation of the crystal. Lets import some EBSD data and computet the grains

mtexdata csl

grains = calcGrains(ebsd);

plot(ebsd,ebsd.orientations)
hold on
plot(grains.boundary)
hold off

We want to consider the following slip systems

sS = slipSystem.fcc(ebsd.CS)
sS = sS.symmetrise;
 
sS = slipSystem  
 mineral: iron (m-3m)
 CRSS: 1
 size: 1 x 1
  u   v   w | h   k   l
  0   1  -1   1   1   1

Since, those slip systems are in crystal coordinates but the stress tensor is in specimen coordinates we either have to rotate the slip systems into specimen coordinates or the stress tensor into crystal coordinates. In the following sections we will demonstrate both ways. Lets start with the first one

% rotate slip systems into specimen coordinates
sSLocal = grains.meanOrientation * sS
 
sSLocal = slipSystem  
 CRSS: 1
 size: 885 x 24

These slip systems are now arranged in matrix form where the rows corrspond to the crystal reference frames of the different grains and the rows are the symmetrically equivalent slip systems. Computing the Schmid faktor we end up with a matrix of the same size

% compute Schmid factor
SF = sSLocal.SchmidFactor(sigma);

% take the maxium allong the rows
[SFMax,active] = max(SF,[],2);

% plot the maximum Schmid factor
plot(grains,SFMax)
mtexColorbar
e = 
  PropertyEvent with properties:

    AffectedObject: [1×1 ColorBar]
            Source: [1×1 matlab.graphics.internal.GraphicsMetaProperty]
         EventName: 'PostSet'

Next we want to visualize the active slip systems.

% take the active slip system and rotate it in specimen coordinates
sSactive = grains.meanOrientation .* sS(active);

hold on
% visualize the trace of the slip plane
quiver(grains,sSactive.trace,'color','b')

% and the slip direction
quiver(grains,sSactive.b,'color','r')
hold off

We observe that the Burgers vector is in most case aligned with the trace. In those cases where trace and Burgers vector are not aligned the slip plane is not perpendicular to the surface and the Burgers vector sticks out of the surface.

Next we want to demonstrate the alternative route

% rotate the stress tensor into crystal coordinates
sigmaLocal = rotate(sigma,inv(grains.meanOrientation))
 
sigmaLocal = stress tensor  
  size   : 885 x 1    
  rank   : 2 (3 x 3)  
  mineral: iron (m-3m)

This becomes a list of stress tensors with respect to crystal coordinates - one for each grain. Now we have both the slip systems as well as the stress tensor in crystal coordiantes and can compute the Schmid factor

% the resulting matrix is the same as above
SF = sS.SchmidFactor(sigmaLocal);

% and hence we may proceed analogously
% take the maxium allong the rows
[SFMax,active] = max(SF,[],2);

% plot the maximum Schmid factor
plot(grains,SFMax)
mtexColorbar

% take the active slip system and rotate it in specimen coordinates
sSactive = grains.meanOrientation .* sS(active);

hold on
% visualize the trace of the slip plane
quiver(grains,sSactive.trace,'color','b')

% and the slip direction
quiver(grains,sSactive.b,'color','r')
hold off
e = 
  PropertyEvent with properties:

    AffectedObject: [1×1 ColorBar]
            Source: [1×1 matlab.graphics.internal.GraphicsMetaProperty]
         EventName: 'PostSet'