Taylor Model edit page

Basic Settings

display pole figure plots with RD on top and ND west

plotx2north

% store old annotation style
storepfA = getMTEXpref('pfAnnotations');

% set new annotation style to display RD and ND
pfAnnotations = @(varargin) text(-[vector3d.X,vector3d.Y],{'RD','ND'},...
  'BackgroundColor','w','tag','axesLabels',varargin{:});

setMTEXpref('pfAnnotations',pfAnnotations);

Slip in Body Centered Cubic Materials

In the following we consider crystallographic slip in bcc materials

% define the slip systems in bcc
cs = crystalSymmetry('432');
sS = slipSystem.bcc(cs)
sS = slipSystem (432)
 size: 1 x 3
 
   u    v    w  | h    k    l CRSS
   1   -1    1    0    1    1    1
  -1    1    1    2    1    1    1
  -1    1    1    3    2    1    1

under plane strain

q = 0;
epsilon = strainTensor(diag([1 -q -(1-q)]))
epsilon = strainTensor (y↑→x)
  type: Lagrange 
  rank: 2 (3 x 3)
 
  1  0  0
  0  0  0
  0  0 -1

The orientation dependence of the Taylor factor

For a family of slip systems sS the Taylor factor M describes the total amount of slip activity that is required to deform a crystal in orientation ori according to strain epsilon. In MTEX this can be computed by the command calcTaylor.

% define a crystal orientation
ori = orientation.byEuler(0,30*degree,15*degree,cs)

% compute the Taylor factor
[M,b,W] = calcTaylor(inv(ori)*epsilon,sS.symmetrise);

M
W
ori = orientation (432 → y↑→x)
 
  Bunge Euler angles in degree
  phi1  Phi phi2
     0   30   15
 
M =
       2.1208
 
W = spinTensor (432)
  rank: 2 (3 x 3)
 
 *10^-2
      0     51  65.65
    -51      0 -23.04
 -65.65  23.04      0

When called without specifying an orientation the command calcTaylor computes the Taylor factor M as well as the spin tensors W as orientation dependent functions, which can be easily visualized and analyzed.

[M,~,W] = calcTaylor(epsilon,sS.symmetrise)

% evaluate the Taylor factor at an arbitrary orientation
M.eval(ori)
W.eval(ori)
M = SO3FunHarmonic (432 → y↑→x)
  bandwidth: 32
  weight: 3.1
 
 
W = SO3VectorFieldHarmonic (1 → y↑→x)
  bandwidth: 32
  tangent space: rightSpinTensor
  intern symmetries: 432 → y↑→x
  intern tangent space: leftVector
 
ans =
       2.1009
 
ans = spinTensor (y↑→x)
  rank: 2 (3 x 3)
 
 *10^-2
      0  38.89  49.68
 -38.89      0 -14.97
 -49.68  14.97      0

The following code reproduces Fig. 5 of the paper of Bunge, H. J. (1970). Some applications of the Taylor theory of polycrystal plasticity. Kristall Und Technik, 5(1), 145-175. http://doi.org/10.1002/crat.19700050112 

% set up an phi1 section plot
sP = phi1Sections(cs);
sP.phi1 = (0:10:90)*degree;

% plot the Taylor factor
plot(M,'smooth',sP)
mtexColorbar

hold on
plot(W,'color','black')
hold off
Warning: Imaginary part of complex valued SO3VectorFields is
ignored. In the following only the real part is plotted.

The orientation dependence of the spin

The norm of the spin tensor is exactly the angle of misorientation a crystal with the corresponding orientation experiences according to Taylor theory. Compare Fig. 8 of the above paper

plot(norm(W)/degree,'smooth',sP,'resolution',0.5*degree)
mtexColorbar

Display the crystallographic spin in sigma sections

sP = sigmaSections(cs);
plot(norm(W)./degree,'smooth',sP)
mtexColorbar

Identification of the most active slip directions

Next we consider a real world data set

mtexdata csl

% compute grains
grains = calcGrains(ebsd('indexed'),'minPixel',3);
grains = smooth(grains,5);
ebsd = EBSD (y↑→x)
 
 Phase   Orientations     Mineral         Color  Symmetry  Crystal reference frame
     0    5 (0.0032%)  notIndexed                                                 
    -1  154107 (100%)        iron  LightSkyBlue      m-3m                         
 
 Properties: ci, error, iq
 Scan unit : um
 X x Y x Z : [0, 511] x [0, 300] x [0, 0]
 Normal vector: (0,0,1)

and apply the Taylor model to each grain of our data set

% some strain
q = 0;
epsilon = strainTensor(diag([1 -q -(1-q)]))

% consider fcc slip systems
sS = symmetrise(slipSystem.fcc(grains.CS));

% apply Taylor model
[M,b,W] = calcTaylor(inv(grains.meanOrientation)*epsilon,sS);
epsilon = strainTensor (y↑→x)
  type: Lagrange 
  rank: 2 (3 x 3)
 
  1  0  0
  0  0  0
  0  0 -1
% colorize grains according to Taylor factor
plot(grains,M)
mtexColorMap white2black
mtexColorbar

% index of the most active slip system - largest b
[~,bMaxId] = max(b,[],2);

% rotate the most active slip system in specimen coordinates
sSGrains = grains.meanOrientation .* sS(bMaxId);

% visualize slip direction and slip plane for each grain
hold on
quiver(grains,sSGrains.b,'autoScaleFactor',0.7,'displayName','Burgers vector','project2plane')
hold on
quiver(grains,sSGrains.trace,'autoScaleFactor',0.7,'displayName','slip plane trace')
hold off

plot the most active slip directions observe that they point all towards the lower hemisphere - why? they do change if q is changed

figure(2)
plot(sSGrains.b)
text([xvector,yvector,zvector],'labeled','BackGroundcolor','w')

Texture evolution during rolling

% define some random orientations
rng(0)
ori = orientation.rand(1e5,grains.CS);

% 30 percent plane strain
q = 0;
epsilon = 0.3 * strainTensor(diag([1 -q -(1-q)]));

numIter = 100;

% compute the Taylor factors and the orientation gradients
[~,~,spin] = calcTaylor(epsilon ./ numIter, sS.symmetrise);

pC = progressCounter(numIter);
for sas = 1:numIter

  % compute the Taylor factors and the orientation gradients
  W = spinTensor(spin.eval(ori).').';

  % rotate the individual orientations
  ori = ori .* orientation(-W);
  pC.show(sas);

end
% plot the resulting pole figures

% set new annotation style to display RD and ND
pfAnnotations = @(varargin) text([vector3d.X,vector3d.Y,vector3d.Z],{'RD','TD','ND'},...
  'BackgroundColor','w','tag','axesLabels',varargin{:});
setMTEXpref('pfAnnotations',pfAnnotations);

plotPDF(ori,Miller({0,0,1},{1,1,1},cs),'contourf')
mtexColorbar

restore MTEX preferences

setMTEXpref('pfAnnotations',storepfA);