The Piezoelectricity Tensor

how to work with piezoelectricity

This m-file mainly demonstrates how to illustrate the directional magnitude of a tensor with mtex

On this page ...
Plotting the magnitude surface
Mean Tensor Calculation

at first, let us import some piezoelectric contents for a quartz specimen.

CS = crystalSymmetry('32', [4.916 4.916 5.4054], 'X||a*', 'Z||c', 'mineral', 'Quartz');

fname = fullfile(mtexDataPath,'tensor', 'Single_RH_quartz_poly.P');

P = loadTensor(fname,CS,'propertyname','piecoelectricity','unit','C/N','interface','P','DoubleConvention')
 
P = tensor  
  propertyname    : piecoelectricity               
  unit            : C/N                            
  rank            : 3 (3 x 3 x 3)                  
  doubleConvention: true                           
  mineral         : Quartz (321, X||a*, Y||b, Z||c)
 
  tensor in compact matrix form:
     0     0     0 -0.67     0   4.6
   2.3  -2.3     0     0  0.67     0
     0     0     0     0     0     0

Plotting the magnitude surface

The default plot of the magnitude, which indicates, in which direction we have the most polarization. By default, we restrict ourselves to the unique region implied by crystal symmetry

% set some colormap well suited for tensor visualisation
setMTEXpref('defaultColorMap',blue2redColorMap);

plot(P)
mtexColorbar
e = 
  PropertyEvent with properties:

    AffectedObject: [1×1 ColorBar]
            Source: [1×1 matlab.graphics.internal.GraphicsMetaProperty]
         EventName: 'PostSet'

but also, we can plot the whole crystal behavior

close all
plot(P,'complete','smooth','upper')
mtexColorbar
e = 
  PropertyEvent with properties:

    AffectedObject: [1×1 ColorBar]
            Source: [1×1 matlab.graphics.internal.GraphicsMetaProperty]
         EventName: 'PostSet'

Most often, the polarization is illustrated as surface magnitude

close all
plot(P,'3d')

Note, that for directions of negative polarization the surface is mapped onto the axis of positive, which then let the surface appear as a double coverage

Quite a famous example in various standard literature is a section through the surface because it can easily be described as an analytical solution. We just specify the plane normal vector

plot(P,'section',zvector)

so we are plotting the polarization in the xy-plane, or the yz-plane with

plot(P,'section',xvector)

Mean Tensor Calculation

Let us import some data, which was originally published by Mainprice, D., Lloyd, G.E. and Casey , M. (1993) Individual orientation measurements in quartz polycrystals: advantages and limitations for texture and petrophysical property determinations. J. of Structural Geology, 15, pp.1169-1187

fname = fullfile(mtexDataPath,'orientation', 'Tongue_Quartzite_Bunge_Euler');

ori = loadOrientation(fname,CS,'interface','generic' ...
  , 'ColumnNames', { 'Euler 1' 'Euler 2' 'Euler 3'}, 'Bunge', 'active rotation')
 
ori = orientation  
  size: 382 x 1
  crystal symmetry : Quartz (321, X||a*, Y||b, Z||c)
  specimen symmetry: 1
 

The figure on p.1184 of the publication

Pm = ori.calcTensor(P)

plot(Pm)
mtexColorbar
 
Pm = tensor  
  propertyname    : piecoelectricity
  rank            : 3 (3 x 3 x 3)   
  doubleConvention: true            
 
  tensor in compact matrix form: *10^-2
 -10.48   34.2 -23.72 -32.75 -64.24 -26.18
 -18.02  -3.15  21.17  62.42  29.67  44.39
 -41.35  40.44   0.91  32.48 -23.42   6.47
e = 
  PropertyEvent with properties:

    AffectedObject: [1×1 ColorBar]
            Source: [1×1 matlab.graphics.internal.GraphicsMetaProperty]
         EventName: 'PostSet'
close all
plot(Pm,'complete','upper')
mtexColorbar

setMTEXpref('defaultColorMap',WhiteJetColorMap)
e = 
  PropertyEvent with properties:

    AffectedObject: [1×1 ColorBar]
            Source: [1×1 matlab.graphics.internal.GraphicsMetaProperty]
         EventName: 'PostSet'