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 = loadEBSD([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
Lets 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 MAD_MAXIMUM= 1.3; % eliminate all meassurements with MAD larger than MAD_MAXIMUM ebsd(ebsd.mad >MAD_MAXIMUM) = [] 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 = tensor(MGlaucophane,csGlaucophane)
CGlaucophane = tensor 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 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 = tensor(MEpidote,csEpidote)
CEpidote = tensor 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 = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: *10^5 25.063 6.002 7.789 -0.235 -0.518 0.617 6.002 17.724 6.209 -0.467 -0.081 0.342 7.789 6.209 23.786 -0.963 -0.325 0.209 -0.235 -0.467 -0.963 7.035 0.267 -0.125 -0.518 -0.081 -0.325 0.267 8.712 -0.2 0.617 0.342 0.209 -0.125 -0.2 7.047 CReuss = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: *10^-3 17.236 4.091 5.131 -0.143 -0.404 0.42 4.091 12.379 4.214 -0.295 -0.059 0.211 5.131 4.214 16.333 -0.631 -0.244 0.13 -0.143 -0.295 -0.631 4.884 0.202 -0.074 -0.404 -0.059 -0.244 0.202 6.131 -0.158 0.42 0.211 0.13 -0.074 -0.158 4.884 CHill = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: *10^5 12.532 3.001 3.894 -0.118 -0.259 0.309 3.001 8.862 3.105 -0.234 -0.041 0.171 3.894 3.105 11.893 -0.481 -0.162 0.104 -0.118 -0.234 -0.481 3.517 0.134 -0.063 -0.259 -0.041 -0.162 0.134 4.356 -0.1 0.309 0.171 0.104 -0.063 -0.1 3.523
for a single phase the syntax is
[CVoigtEpidote,CReussEpidote,CHillEpidote] = calcTensor(ebsd('Epidote'),CEpidote)
CVoigtEpidote = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: *10^4 94.81 25.42 30.87 -0.67 -1.19 1.92 25.42 80.01 25.77 0.01 0.34 -0.05 30.87 25.77 92.85 -0.81 -0.04 0.56 -0.67 0.01 -0.81 26.71 0.7 0.24 -1.19 0.34 -0.04 0.7 32.85 -0.36 1.92 -0.05 0.56 0.24 -0.36 26.84 CReussEpidote = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: *10^-3 43.69 12.79 15.43 -0.37 -0.52 0.92 12.79 35.77 13.14 -0.02 0.27 0.01 15.43 13.14 42.89 -0.39 0.17 0.23 -0.37 -0.02 -0.39 11.46 0.37 0.1 -0.52 0.27 0.17 0.37 14.74 -0.16 0.92 0.01 0.23 0.1 -0.16 11.53 CHillEpidote = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: *10^4 47.4 12.71 15.43 -0.34 -0.6 0.96 12.71 40 12.89 0 0.17 -0.03 15.43 12.89 46.42 -0.41 -0.02 0.28 -0.34 0 -0.41 13.35 0.35 0.12 -0.6 0.17 -0.02 0.35 16.42 -0.18 0.96 -0.03 0.28 0.12 -0.18 13.42
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 Portion specified by Fourier coefficients: degree: 28 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 = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: 207.41 60.93 65.68 -0.81 -0.85 2.25 60.93 170.74 61.02 -1.13 -0.14 2.25 65.68 61.02 203.97 -1.13 -0.85 0.56 -0.81 -1.13 -1.13 63.17 0.85 -0.35 -0.85 -0.14 -0.85 0.85 70 -0.16 2.25 2.25 0.56 -0.35 -0.16 64.07 CReussEpidote = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: 192.78 62.17 65.86 -1.02 -0.07 2.27 62.17 157.03 62.9 -1.03 0.29 1.92 65.86 62.9 190.66 -1.17 -0.06 0.3 -1.02 -1.03 -1.17 54.84 0.97 -0.18 -0.07 0.29 -0.06 0.97 62.92 -0.08 2.27 1.92 0.3 -0.18 -0.08 55.66 CHillEpidote = tensor rank: 4 (3 x 3 x 3 x 3) tensor in Voigt matrix representation: 200.09 61.55 65.77 -0.91 -0.46 2.26 61.55 163.88 61.96 -1.08 0.07 2.08 65.77 61.96 197.31 -1.15 -0.46 0.43 -0.91 -1.08 -1.15 59 0.91 -0.27 -0.46 0.07 -0.46 0.91 66.46 -0.12 2.26 2.08 0.43 -0.27 -0.12 59.86
MTEX 4.0.10 |