Tensor Visualization edit page

A tensor of rank two or higher has too many components to be read off a list. What can be looked at instead is its directional magnitude - the scalar that the tensor produces when it is contracted with one and the same direction in all of its slots. For a tensor \(T\) of rank \(r\) this is

\[ Q(\vec x) = T_{i_1 \ldots i_r}\, x_{i_1} \cdots x_{i_r}, \qquad \vec x = 1, \]

a function on the sphere, i.e. an S2Fun, and plotting it is what the plot command of a tensor does by default.

setMTEXpref('defaultColorMap',blue2redColorMap);

cs = crystalSymmetry('mmm',[4.7646 10.2296 5.9942],'mineral','Olivine');
C = stiffnessTensor.load(fullfile(mtexDataPath,'tensor','Olivine1997PC.GPa'),cs)
C = stiffnessTensor (Olivine)
  unit: GPa              
  rank: 4 (3 x 3 x 3 x 3)
 
  tensor in Voigt matrix representation:
 320.5  68.2  71.6     0     0     0
  68.2 196.5  76.8     0     0     0
  71.6  76.8 233.5     0     0     0
     0     0     0    64     0     0
     0     0     0     0    77     0
     0     0     0     0     0  78.7
plot(C,'complete','upper')

mtexColorbar

In particular, all of the spherical plotting options apply - filled contours, contour lines, 3d plots, as well as any spherical projection.

Furthermore, we can access this directional function directly by

sF = C.directionalMagnitude
sF = S2FunHarmonicSym (Olivine)
  bandwidth: 4
  antipodal: true

and use it to derive e.g. the location of the maximum directional magnitude

[v,pos] = max(sF)
v =
  320.5000
 
pos = Miller (Olivine)
 antipodal: true
       h      k      l
  4.7646 0.0013 0.0011

Rank two tensors and their principal axes

For a symmetric rank two tensor the directional magnitude is the quadratic form \(\vec x^T T \vec x\), and its extrema are attained along the eigenvectors. Hence the plot shows the principal axes directly.

T = tensor(diag([3 1 -1]),'rank',2,plottingConvention('y↑→x'))
T = tensor (y↑→x)
  rank: 2 (3 x 3)
 
  3  0  0
  0  1  0
  0  0 -1
plot(T,'complete','upper')
mtexColorbar

the principle axes of a symmetric rank two tensor are computed by the command eig 

[e,lambda] = eig(T);

hold on
plot(e,'plane','linewidth',2,'antipodal')
hold off

Sections

Some derived quantities are not functions of a single direction. The Poisson ratio and the shear modulus, for instance, depend on two directions, and it is only meaningful to evaluate them for directions perpendicular to the first one. Such functions are best drawn as a section along the corresponding great circle - see the elasticity chapter.

p = vector3d.Z;

plotSection(C.PoissonRatio(p),p,'color','interp','linewidth',5)
axis off
mtexColorbar

Specialized plots

Beyond the directional magnitude there are visualizations that only make sense for one particular kind of tensor - wave velocities and their polarizations for the elasticity tensor, the birefringence of the refractive index tensor, or the piezoelectric modulus. Each is described in its own chapter.

setMTEXpref('defaultColorMap',WhiteJetColorMap)