# Average Material Tensors

how to calculate average material tensors from ODF and EBSD data

MTEX offers several ways to compute average material tensors from ODFs or EBSD data.

 On this page ... Import EBSD Data Data Correction Define Elastic Stiffness Tensors for Glaucophane and Epidote The Average Tensor from EBSD Data ODF Estimation The Average Tensor from an ODF

set up a nice colormap

`setMTEXpref('defaultColorMap',blue2redColorMap);`

## Import EBSD Data

We start by importing some EBSD data of Glaucophane and Epidote.

```ebsd = EBSD.load([mtexDataPath '/EBSD/data.ctf'],...
'convertEuler2SpatialReferenceFrame')```
```
ebsd = EBSD

Phase  Orientations      Mineral        Color  Symmetry  Crystal reference frame
0   28015 (56%)   notIndexed
1   13855 (28%)  Glaucophane   light blue     12/m1       X||a*, Y||b*, Z||c
2   4603 (9.2%)      Epidote  light green     12/m1       X||a*, Y||b*, Z||c
3   3213 (6.4%)       Pyrope    light red      m-3m
4   295 (0.59%)    omphacite         cyan     12/m1       X||a*, Y||b*, Z||c

Properties: bands, bc, bs, error, mad, x, y
Scan unit : um

```

Let's visualize a subset of the data

`plot(ebsd(inpolygon(ebsd,[2000 0 1400 375])))`

## Data Correction

next, we correct the data by excluding orientations with large MAD value

```% define maximum acceptable MAD value

plot(ebsd(inpolygon(ebsd,[2000 0 1400 375])))```
```
ebsd = EBSD

Phase  Orientations      Mineral        Color  Symmetry  Crystal reference frame
0   28015 (56%)   notIndexed
1   13779 (28%)  Glaucophane   light blue     12/m1       X||a*, Y||b*, Z||c
2   4510 (9.1%)      Epidote  light green     12/m1       X||a*, Y||b*, Z||c
3   3212 (6.5%)       Pyrope    light red      m-3m
4   218 (0.44%)    omphacite         cyan     12/m1       X||a*, Y||b*, Z||c

Properties: bands, bc, bs, error, mad, x, y
Scan unit : um

```

## Define Elastic Stiffness Tensors for Glaucophane and Epidote

Glaucophane elastic stiffness (Cij) Tensor in GPa Bezacier, L., Reynard, B., Bass, J.D., Wang, J., and Mainprice, D. (2010) Elasticity of glaucophane and seismic properties of high-pressure low-temperature oceanic rocks in subduction zones. Tectonophysics, 494, 201-210.

```% define the tensor coefficients
MGlaucophane =....
[[122.28   45.69   37.24   0.00   2.35   0.00];...
[  45.69  231.50   74.91   0.00  -4.78   0.00];...
[  37.24   74.91  254.57   0.00 -23.74   0.00];...
[   0.00    0.00    0.00  79.67   0.00   8.89];...
[   2.35   -4.78  -23.74   0.00  52.82   0.00];...
[   0.00    0.00    0.00   8.89   0.00  51.24]];

% define the reference frame
csGlaucophane = crystalSymmetry('2/m',[9.5334,17.7347,5.3008],...
[90.00,103.597,90.00]*degree,'X||a*','Z||c','mineral','Glaucophane');

% define the tensor
CGlaucophane = stiffnessTensor(MGlaucophane,csGlaucophane)```
```
CGlaucophane = stiffnessTensor
unit   : GPa
rank   : 4 (3 x 3 x 3 x 3)
mineral: Glaucophane (12/m1, X||a*, Y||b*, Z||c)

tensor in Voigt matrix representation:
122.28  45.69  37.24      0   2.35      0
45.69  231.5  74.91      0  -4.78      0
37.24  74.91 254.57      0 -23.74      0
0      0      0  79.67      0   8.89
2.35  -4.78 -23.74      0  52.82      0
0      0      0   8.89      0  51.24
```

Epidote elastic stiffness (Cij) Tensor in GPa Aleksandrov, K.S., Alchikov, U.V., Belikov, B.P., Zaslavskii, B.I. and Krupnyi, A.I.: 1974 'Velocities of elastic waves in minerals at atmospheric pressure and increasing the precision of elastic constants by means of EVM (in Russian)', Izv. Acad. Sci. USSR, Geol. Ser.10, 15-24.

```% define the tensor coefficients
MEpidote =....
[[211.50    65.60    43.20     0.00     -6.50     0.00];...
[  65.60   239.00    43.60     0.00    -10.40     0.00];...
[  43.20    43.60   202.10     0.00    -20.00     0.00];...
[   0.00     0.00     0.00    39.10      0.00    -2.30];...
[  -6.50   -10.40   -20.00     0.00     43.40     0.00];...
[   0.00     0.00     0.00    -2.30      0.00    79.50]];

% define the reference frame
csEpidote= crystalSymmetry('2/m',[8.8877,5.6275,10.1517],...
[90.00,115.383,90.00]*degree,'X||a*','Z||c','mineral','Epidote');

% define the tensor
CEpidote = stiffnessTensor(MEpidote,csEpidote)```
```
CEpidote = stiffnessTensor
unit   : GPa
rank   : 4 (3 x 3 x 3 x 3)
mineral: Epidote (12/m1, X||a*, Y||b*, Z||c)

tensor in Voigt matrix representation:
211.5  65.6  43.2     0  -6.5     0
65.6   239  43.6     0 -10.4     0
43.2  43.6 202.1     0   -20     0
0     0     0  39.1     0  -2.3
-6.5 -10.4   -20     0  43.4     0
0     0     0  -2.3     0  79.5
```

## The Average Tensor from EBSD Data

The Voigt, Reuss, and Hill averages for all phases are computed by

`[CVoigt,CReuss,CHill] =  calcTensor(ebsd({'Epidote','Glaucophane'}),CGlaucophane,CEpidote)`
```
CVoigt = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
216.77  52.91  67.88  -1.96   -4.2   5.19
52.91 158.05  54.54  -3.39  -0.46   2.47
67.88  54.54 206.78  -7.29  -2.37   1.72
-1.96  -3.39  -7.29  60.88    2.2  -0.82
-4.2  -0.46  -2.37    2.2  75.31  -1.58
5.19   2.47   1.72  -0.82  -1.58  61.01

CReuss = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
197.74  48.59  60.64  -1.71  -4.26   4.72
48.59 145.04  49.96  -2.86  -0.38   2.02
60.64  49.96 188.42   -6.3   -2.2   1.41
-1.71  -2.86   -6.3  55.31   2.18   -0.6
-4.26  -0.38   -2.2   2.18  69.75  -1.61
4.72   2.02   1.41   -0.6  -1.61  55.39

CHill = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
207.25  50.75  64.26  -1.83  -4.23   4.96
50.75 151.55  52.25  -3.13  -0.42   2.24
64.26  52.25  197.6  -6.79  -2.28   1.57
-1.83  -3.13  -6.79  58.09   2.19  -0.71
-4.23  -0.42  -2.28   2.19  72.53  -1.59
4.96   2.24   1.57  -0.71  -1.59   58.2
```

for a single phase the syntax is

`[CVoigtEpidote,CReussEpidote,CHillEpidote] =  calcTensor(ebsd('Epidote'),CEpidote)`
```
CVoigtEpidote = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
210.22  56.37  68.44  -1.49  -2.64   4.27
56.37  177.4  57.14   0.01   0.75  -0.11
68.44  57.14 205.87   -1.8  -0.08   1.24
-1.49   0.01   -1.8  59.21   1.55   0.53
-2.64   0.75  -0.08   1.55  72.83  -0.79
4.27  -0.11   1.24   0.53  -0.79  59.51

CReussEpidote = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
197.04  57.67  69.57  -1.69  -2.32   4.16
57.67 161.32  59.28  -0.11   1.21   0.03
69.57  59.28 193.44  -1.78   0.78   1.05
-1.69  -0.11  -1.78   51.7   1.69   0.44
-2.32   1.21   0.78   1.69  66.46  -0.71
4.16   0.03   1.05   0.44  -0.71  51.98

CHillEpidote = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
203.63  57.02     69  -1.59  -2.48   4.22
57.02 169.36  58.21  -0.05   0.98  -0.04
69  58.21 199.66  -1.79   0.35   1.14
-1.59  -0.05  -1.79  55.46   1.62   0.48
-2.48   0.98   0.35   1.62  69.65  -0.75
4.22  -0.04   1.14   0.48  -0.75  55.74
```

## ODF Estimation

Next, we estimate an ODF for the Epidote phase

`odfEpidote = calcODF(ebsd('Epidote').orientations,'halfwidth',10*degree)`
```
odfEpidote = ODF
crystal symmetry : Epidote (12/m1, X||a*, Y||b*, Z||c)
specimen symmetry: 1

Harmonic portion:
degree: 25
weight: 1

```

## The Average Tensor from an ODF

The Voigt, Reuss, and Hill averages for the above ODF are computed by

```[CVoigtEpidote, CReussEpidote, CHillEpidote] =  ...
calcTensor(odfEpidote,CEpidote)

% set back the colormap
setMTEXpref('defaultColorMap',WhiteJetColorMap);```
```
CVoigtEpidote = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
208.94  57.35  67.73  -1.31  -2.25   3.75
57.35 177.36  57.99  -0.15   0.58   0.23
67.73  57.99 204.96  -1.61  -0.18   1.08
-1.31  -0.15  -1.61  60.13   1.37   0.37
-2.25   0.58  -0.18   1.37  72.05  -0.66
3.75   0.23   1.08   0.37  -0.66  60.49

CReussEpidote = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
195.18  58.63  68.58  -1.49  -1.82   3.66
58.63 161.47     60  -0.24   0.99    0.3
68.58     60 192.09   -1.6   0.62   0.87
-1.49  -0.24   -1.6   52.4   1.49   0.35
-1.82   0.99   0.62   1.49  65.38  -0.57
3.66    0.3   0.87   0.35  -0.57  52.74

CHillEpidote = stiffnessTensor
unit: GPa
rank: 4 (3 x 3 x 3 x 3)

tensor in Voigt matrix representation:
202.06  57.99  68.15   -1.4  -2.03    3.7
57.99 169.42  58.99  -0.19   0.79   0.27
68.15  58.99 198.53   -1.6   0.22   0.97
-1.4  -0.19   -1.6  56.27   1.43   0.36
-2.03   0.79   0.22   1.43  68.71  -0.62
3.7   0.27   0.97   0.36  -0.62  56.61
```