2nd rank tensor using the harmonic method and plot with surf

[David Mainprice]

Please be as detailed as possible, explain what you want to do, what kind of data you are using. Attaching simplified code and data is always useful. Click the the button { } to inline code and finally remove these lines before posting :)

It is my understanding that 2nd rank tensor with positive values at T11=T22=6.5 and T33=11.3 like thermal conductivirty of alpha-quartz can be represented by an ellipsoid.

I tied to make an example of script of this.  Posted below with plot(TC_harmonic) I get peanut shaped plot in 3D.

I assumed the TC_harmonic = thermal_conductivity.directionalMagnitude

TC_harmonic = S2FunHarmonic (show methods, plot)

mineral: Alpha-Quartz (-3m1, X||a, Y||b*, Z||c*)

bandwidth: 4

antipodal: true

It is possible I made a mistake some where !

all the best David

%% Clear memory and close all plots

clear all;

close all;

%% Matrix of alpha-quartz thermal conductivity values in (W/m K)

% x = a-axis y = m-axis z = c-axis

M = [[  6.5   0.0   0.0 ]; ...

    [   0.0   6.5   0.0 ]; ...

    [   0.0   0.0  11.3 ]];

% Define tenor object in MTEX thermal conductivity tensor

cs_tensor = crystalSymmetry('-3m', [4.1934 4.1934 5.4052],...

    'X||a', 'Z||c', 'mineral', 'Alpha-Quartz')

thermal_conductivity = tensor(M,cs_tensor,'propertyname',...

    'Thermal conductivity','units','W/m K')

%**************************************************************************

%% Pole figure representation of thermal conductivity values

%**************************************************************************

% [111] = 8.1 W/m K

directionalMagnitude(thermal_conductivity,vector3d(1,1,1))

% [2-1-10]= 6.5 W/m K

directionalMagnitude(thermal_conductivity,Miller(2,-1,-1,0,cs_tensor,'UVTW'))

% [2-1-10]= 11.3 W/m K

directionalMagnitude(thermal_conductivity,Miller(0,0,0,1,cs_tensor,'UVTW'))

figure

plota2east

plot(thermal_conductivity,'complete','upper')

mtexColorMap white2black

mtexColorbar('Title','W/m K','FontSize',25,'location','southoutside')

annotate([Miller(2,-1,-1,0,cs_tensor,'UVTW'),...

Miller(0,1,-1,0,cs_tensor,'UVTW'),...

Miller(0,0,0,1,cs_tensor,'UVTW')],'all','labeled','BackgroundColor','w',...

'MarkerSize',20,'FontSize',30)

drawNow(gcm,'figSize','medium')

saveFigure('Example_Thermal_Conductivity.pdf')

%**************************************************************************

%% The harmonic method for radius vector with value of thermal conductivity 

%**************************************************************************

% generate harmonic coefficients

TC_harmonic = thermal_conductivity.directionalMagnitude

figure

surf(TC_harmonic)

mtexColorMap white2black

axis equal

axis tight

drawNow(gcm,'figSize','medium')

saveFigure('Example_Thermal_Conductivity_surf.pdf')

%%

% TC_harmonic = S2FunHarmonic (show methods, plot)

% mineral: Alpha-Quartz (-3m1, X||a, Y||b*, Z||c*)

% bandwidth: 4

% antipodal: true

figure ; plot(TC_harmonic,'complete','upper')

[ruediger Kilian]

Hi David,

if you want an ellipsoid with the eigenvalues as axes, you could try the following:

M = [[ 6.5 0.0 0.0 ]; ...
[ 0.0 6.5 0.0 ]; ...
[ 0.0 0.0 11.3 ]];

cs_tensor = crystalSymmetry('-3m', [4.1934 4.1934 5.4052],...
'X||a', 'Z||c', 'mineral', 'Alpha-Quartz')

tc = tensor(M,cs_tensor,'propertyname',...

'Thermal conductivity','units','W/m K')

% directional magnitude
surf(tc.directionalMagnitude)
nextAxis
% principal axes
sp=eig(tc)
[x, y, z] = ellipsoid(0,0,0,sp(1),sp(2),sp(3),50);
c=norm(vector3d(x,y,z));
surf(x, y, z, c)
colorbar
axis equal

Cheers,
Rüdiger

[Ralf Hielscher]

Hi David and Rüdiger,

I think it is important to distinguish the following two plots

1. the surface described by direction multiplied by property value - this is the result of the surface plot

2. the isosurface, i.e., the surface of vectors v such that v * property value = const - this gives indeed an ellipse but with axislength inverse to the property values.

I do not know what is more common.

Ralf.

[David Mainprice]

Dear All,

Given the 2nd rank symmetry tensor has for positive values the traditional  an ellipsoid as the expected  3D representation is the ellipsoid.

From Nye’s book - (1) magnitude  sigma (2) unit sphere (3) quadric 1/ square root(sigma)

for velocities it is probable magnitude velocity and direction 

I hope this helps

all the best David

Le 12 nov. 2018 à 16:56, Ralf Hielscher <ralf.hi...@gmail.com> a écrit :.

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