Transformation Texture edit page

Transformation Texture

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 → xyz)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  48.4641 88.1693 158.931

is transformed in one of the following Ferrite orientations

oriB = variants(p2c,oriA)

oriB = orientation (Ferrite → xyz)
  size: 1 x 12
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  88.1838 112.697  342.55
  329.873 113.216 96.2497
  56.3001 42.8593 249.448
  292.141 26.4907 294.641
   182.75  96.601 193.475
   79.533 52.3014 53.6058
  187.686 115.492 14.8591
  41.1175 133.511 70.1362
  128.625  66.109 272.324
  18.7476  124.54 235.899
  337.793 25.0672 73.1235
   93.935 94.0384 163.898

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 → xyz)
 
  <strong>unimodal component</strong>
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 1 orientations
 
  Bunge Euler angles in degree
     phi1     Phi    phi2  weight
  48.4641 88.1693 158.931       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 → xyz)
  size: 10000 x 1
 
oriBSim = orientation (Ferrite → xyz)
  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 → xyz)
  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 → xyz)
  bandwidth: 25
  weight: 1
 
ans =
    3.4889

% texture index of the directly computed transformed ODF
norm(odfB)^2

ans =
   17.4509