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Figures

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Subfigures

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Tables

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Equations

Below is an example of a equation:

\[\int_{\Omega} \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} d\Omega = 0 \label{eq:sigma}\tag{1}\]

\[\boldsymbol{\sigma} = \mathbb{C} : \boldsymbol{\varepsilon} \label{eq:sigma_c_epsilon}\tag{2}\]

Stress \(\boldsymbol{\sigma}\) in Eq.\(\ref{eq:sigma}\) is defined by Eq.\(\ref{eq:sigma_c_epsilon}\). Numbering of equations in html does not seem to work nicely.

\[\begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{yz} \\ \sigma_{zx} \\ \sigma_{xy} \end{bmatrix} = \begin{bmatrix} D_{11} & D_{12} & D_{12} & 0 & 0 & 0 \\ D_{12} & D_{11} & D_{12} & 0 & 0 & 0 \\ D_{12} & D_{12} & D_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & D_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & D_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & D_{44} \end{bmatrix} \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ \varepsilon_{zz} \\ \gamma_{yz} \\ \gamma_{zx} \\ \gamma_{xy} \end{bmatrix} \label{}\tag{3}\]

References

This is my recent publication (Mikula, Vastola, and Zhang 2024).

Large portion of code

% PURPOSE: 
% Example of tensor rotation about a general axis
% Rotating tensor of elastic constants about an arbitrary vector o
%
% INPUT:
%   > elastic constants D_11, D_12, D_44 
%   > rotation axis o1, o2, o3 
%   > rotation angle theta 
% OUTPUT:
%   > rotated tensor of elastic constants in matrix form D_rot

% -----------------------------------------------------------

% Material : [GPa]
    D_11 = 190
    D_12 = 161
%D_44 = (D_11-D_12)/2 % uncomment for isotropic elasticity
    D_44 = 42.3

% Axis to rotate about
% Make sure that |o|=1
    o1 = 0
    o2 = 0
    o3 = 1

% Specify the angle [RAD] 
    theta = -pi/4

% -----------------------------------------------------------

% Tensor of elastic constants (cubic elasticity) in the matrix form
% Voigth notation
D = [D_11 D_12 D_12 0 0 0 
     D_12 D_11 D_12 0 0 0
     D_12 D_12 D_11 0 0 0
     0 0 0 D_44 0 0
     0 0 0 0 D_44 0
     0 0 0 0 0 D_44];

% Matrix due to the conversion from engineering to tensorial strain  
R(:,:) = 0.0d0;
R(1,1) = 1.0d0;
R(2,2) = 1.0d0;
R(3,3) = 1.0d0;
R(4,4) = 2.0d0;
R(5,5) = 2.0d0;
R(6,6) = 2.0d0;

invR(:,:) = R(:,:);
invR(4,4) = .5d0;
invR(5,5) = .5d0;
invR(6,6) = .5d0;


% Rotating about an arbitrary axis (between crystal and global coordinate system)
R_cg=[o1^2+(1-o1^2)*cos(theta) o1*o2*(1-cos(theta))-o3*sin(theta) o1*o3*(1-cos(theta))+o2*sin(theta)
   o1*o2*(1-cos(theta))+o3*sin(theta) o2^2+(1-o2^2)*cos(theta) o2*o3*(1-cos(theta))-o1*sin(theta)
   o1*o3*(1-cos(theta))-o2*sin(theta) o2*o3*(1-cos(theta))+o1*sin(theta) o3^2+(1-o3^2)*cos(theta)];


for i=1:3
for j=1:3

mmod = [1 1 1 1 1
        -1 2 2 2 2
        0 0 3 3 3
        1 1 1 4 4
        2 2 2 2 5];

 K1(i,j) = R_cg(i,j)^2;
 K2(i,j) = R_cg(i,mmod(j+1,3))*R_cg(i,mmod(j+2,3));
 K3(i,j) = R_cg(mmod(i+1,3),j)*R_cg(mmod(i+2,3),j);
 K4(i,j) = R_cg(mmod(i+1,3),mmod(j+1,3))*R_cg(mmod(i+2,3),mmod(j+2,3)) + ...
           R_cg(mmod(i+1,3),mmod(j+2,3))*R_cg(mmod(i+2,3),mmod(j+1,3));
end
end

T_cg(1:3,1:3) = K1;
T_cg(1:3,4:6) = 2.0d0*K2;
T_cg(4:6,1:3) = K3;
T_cg(4:6,4:6) = K4;

invT_cg(1:3,1:3) = K1';
invT_cg(1:3,4:6) = 2.0d0*K3';
invT_cg(4:6,1:3) = K2';
invT_cg(4:6,4:6) = K4';

% Rotated tensor of elastic constants
D_rot = T_cg*D*R*invT_cg*invR

Test reference to Eq.\(\ref{eq:sigma_c_epsilon}\).

References