Transformation Texture edit page

During phase transformation or twinning the orientation of a crystal rapidly flips from an initial state oriA into a transformed state oriB. This relationship between the initial and transformed state can be described by an orientation relationship OR. To make the situation more precise, we consider the phase transformation from austenite to ferrite via the Nishiyama Wassermann orientation relationship

% parent and child crystal symmetry
csP = crystalSymmetry('432','mineral','Austenite');
csC = crystalSymmetry('432','mineral','Ferrite');

% the orientation relationship
p2c = orientation.NishiyamaWassermann(csP,csC);

Now an arbitrary Austenite orientation

oriA = orientation.rand(csP)
oriA = orientation (Austenite → y←↑x)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  156.958 161.468 197.878

is transformed in one of the following Ferrite orientations

oriB = variants(p2c,oriA)
oriB = orientation (Ferrite → y←↑x)
  size: 1 x 12
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  95.6328 91.0043 16.4349
  330.335 152.635 96.6679
  23.6163  57.205 298.401
  311.882  62.194 268.771
  186.966 63.9508 170.719
  46.1511 49.9141 104.876
  183.862 83.1959 351.609
  80.4228 132.659 125.129
   110.65 31.2299 298.288
  54.5999 141.255 286.055
  333.585 64.3114 78.9078
  89.9532 72.3537 197.269

These 12 Ferrite orientations are called variants of the orientation relationship. Lets visualize them in a pole figure plot

hC = Miller({1,1,1},{1,1,0},csC);
hP = Miller({1,1,0},{1,0,0},csP);

% plot the child variants
plotPDF(oriB,hC,'MarkerSize',5,'markerColor','black','figSize','medium');

% and on top the parent orientation
opt = {'MarkerFaceColor','none','MarkerEdgeColor','darkred','linewidth',3};
for k = 1:2
  nextAxis(k)
  hold on
  plot(oriA * hP(k).symmetrise ,opt{:})
  xlabel(char(hP(k),'latex'),'Color','red','Interpreter','latex')
  hold off
end
drawNow(gcm)

In case we have multiple parent orientations following some initial orientation distribution function odf 

% define a model ODF
odfA = unimodalODF(oriA,'halfwidth',5*degree)

plotPDF(odfA,hP,'figSize','medium')
mtexColorbar
odfA = SO3FunRBF (Austenite → y←↑x)
 
  unimodal component
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 1 orientations
 
  Bunge Euler angles in degree
     phi1     Phi    phi2  weight
  156.958 161.468 197.878       1

We can draw some random orientations according this model ODF and apply the same commands variants to compute all transformed orientations in one step

% number of discrete orientations
n = 10000;
oriASim = odfA.discreteSample(n)

% transform the orientations
oriBSim = variants(p2c,oriASim)

% show the result
plotPDF(oriBSim,hC,'contourf','figSize','medium');
mtexColorbar
oriASim = orientation (Austenite → y←↑x)
  size: 10000 x 1
 
oriBSim = orientation (Ferrite → y←↑x)
  size: 10000 x 12

An alternative and better approach is to directly use odfA as an input to the function variants. In this case the output is the orientation distribution function of the transformed material

% compute the child ODF
odfB = variants(p2c,odfA)

% plot
plotPDF(odfB,hC,'contourf','figSize','medium');
mtexColorbar
odfB = SO3FunHarmonic (Ferrite → y↑→x)
  bandwidth: 48
  weight: 1

We observe that the transformed ODF computed by the latter approach is sharper and shows more details when compared with the ODF computed from discrete orientations. We may quantify this difference by computing the texture index of both ODFs

% texture index of the transformed ODF computed from discrete orientations
odfBSim = calcDensity(oriBSim)
norm(odfBSim)^2
odfBSim = SO3FunHarmonic (Ferrite → y←↑x)
  bandwidth: 25
  weight: 1
 
ans =
    3.4854
% texture index of the directly computed transformed ODF
norm(odfB)^2
ans =
   17.4509