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# Euler Angles Bunge vs Elastic

### MATLAB code @github private repository

The MATLAB code outputs the sample Normal direction (Rolling direction, Transverse direction) in the grain frame. ‘main_Euler_Angles_Bunge.m’ works as ‘API’ in which you can input EBSD grain number, Bunge. The functions are defined in other m files. Finally, it outputs Normal direction (Rolling direction, Transverse direction) in the grain frame and phi & theta for elastic modulus in one excel file.

### MATLAB code @github private repository

The MATLAB code outputs the sample normal direction in the grain frame. ‘main_Euler_Angles_Elastic.m’ works as ‘API’. The functions are defined in other m files. Finally, it outputs normal direction in the grain frame and phi & theta vs elastic modulus in one excel file.

### EBSD Euler Angles Bunge (phi1, theta, phi2) -> XYZ -> phi, theta for Elastic

theta in Bunge is different from theta for Elastic, please refer to the description at 2 sections: Euler Angle Bunge and Normal direction in the Grain frame

### Notation

$\phi_1&space;\texttt{:the&space;rotation,&space;the&space;sample&space;Z&space;axis}.$

$\theta&space;\texttt{:the&space;rotation,&space;the&space;grain&space;x&space;axis}.$

$\phi_2&space;\texttt{:the&space;rotation,&space;the&space;grain&space;z&space;axis}.$

$R=\begin{bmatrix}&space;cos&space;\phi_2&space;&&space;sin\phi_2&space;&&space;0\\&space;-sin\phi_2&space;&&space;cos\phi_2&space;&&space;0\\&space;0&space;&&space;0&space;&&space;1&space;\end{bmatrix}&space;\begin{bmatrix}&space;1&space;&&space;0&space;&&space;0\\&space;0&space;&&space;cos\theta&space;&&space;sin\theta\\&space;0&space;&&space;-sin\theta&space;&&space;cos\theta&space;\end{bmatrix}&space;\begin{bmatrix}&space;cos\phi_1&space;&&space;sin\phi_1&space;&&space;0\\&space;-sin\phi_1&space;&&space;cos\phi_1&space;&&space;0\\&space;0&space;&&space;0&space;&&space;1&space;\end{bmatrix}$

$R=\begin{bmatrix}&space;cos\phi_1&space;cos\phi_2&space;-cos\theta&space;sin\phi_1&space;sin&space;\phi_2&space;&&space;sin\phi_1&space;cos\phi_2&space;+&space;cos\theta&space;cos\phi_1&space;sin\phi_2&space;&&space;sin\theta&space;sin\phi_2\\&space;-cos\phi_1&space;sin\phi_2&space;-cos\theta&space;sin\phi_1&space;cos&space;\phi_2&space;&&space;-sin\phi_1&space;sin\phi_2&space;+&space;cos\theta&space;cos\phi_1&space;cos&space;\phi_2&space;&&space;sin\theta&space;cos\phi_2\\&space;sin\theta&space;sin\phi_1&space;&&space;-sin\theta&space;cos\phi_1&space;&&space;cos\theta&space;\end{bmatrix}$

${\color{Blue}&space;R^{-1}}=\begin{bmatrix}&space;cos\phi_1&space;cos\phi_2&space;-cos\theta&space;sin\phi_1&space;sin&space;\phi_2&space;&&space;-cos\phi_1&space;sin\phi_2&space;-&space;cos\theta&space;sin\phi_1&space;cos\phi_2&space;&&space;sin\theta&space;sin\phi_1\\&space;sin\phi_1&space;cos\phi_2&space;+cos\theta&space;cos\phi_1&space;sin&space;\phi_2&space;&&space;-sin\phi_1&space;sin\phi_2&space;+&space;cos\theta&space;cos\phi_1&space;cos&space;\phi_2&space;&&space;-sin\theta&space;cos\phi_1\\&space;sin\theta&space;sin\phi_2&space;&&space;-sin\theta&space;cos\phi_2&space;&&space;cos\theta&space;\end{bmatrix}$

First of all, the sample frame coincides with one grain frame.

R is based on the sample frame. Normal direction (Z) cosine 001 in the sample frame, R * Z -> one grain 001 orientation (Z’) in the sample frame.

Inverse of R is based on the grain frame. One grain 001 orientation (z) in the grain frame, R-1 * z -> sample normal direction in the grain frame.

### Sample frame Normal direction, Rolling direction and Transverse direction

https://en.wikipedia.org/wiki/Euler_angles

http://solidmechanics.org/text/Chapter3_2/Chapter3_2.htm

http://progs.coudert.name/elate

### Input: stiffness matrix (coefficients in GPa) of Cu

     168.4    121.4    121.4        0        0        0
121.4    168.4    121.4        0        0        0
121.4    121.4    168.4        0        0        0
0        0        0     75.4        0        0
0        0        0        0     75.4        0
0        0        0        0        0     75.4

https://www.wolframcloud.com/objects/zp2130/Published/IPF_Color_Map.nb

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