Euler Angles Bunge vs Elastic

Euler Angles Bunge

MATLAB code @github private repository

Euler Angles 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.

Verify progs.coudert.name/elate

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

phi,theta for MATLAB plotting Orientation vs Elastic -> Angle wrt xyz@grain frame -> phi,theta for Searching Elastic -> Elastic

Grain 1-4

Euler Angle Bunge

Notation

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

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

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

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

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

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

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.

Normal direction in the Grain frame

Sample frame Normal direction 001

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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Euler Angles Bunge vs Elastic