User:Eas4200c.f08.WIKI.E/extra2

This is equation one which will be combined with two later equations to give us the generalized 3-D case:

$$t_y=\sigma_{yy}n_y+\sigma_{yz}n_z\;$$

Note: $$\left[t_y \right]=\frac{F}{L^2}$$

$$\vec{t}\;$$:traction vector (distribution surface force)

$$\left[t_y \right]=\left[\sigma \right]$$

This is equation 2:

$$t_z=\sigma _{yz}\cdot n_y+\sigma _{zz}n_z$$

This is equations one and two combined into matrix format:

$$\begin{Bmatrix} t_y\\ t_z \end{Bmatrix}=\begin{bmatrix} \sigma_{yy} &\sigma_{yz} \\ \sigma_{zy}& \sigma_{zz} \end{bmatrix}\begin{Bmatrix} n_y\\ n_z \end{Bmatrix}$$

Generalized to 3-D case

$$\begin{Bmatrix} t_1\\ t_2\\ t_3 \end{Bmatrix}=\begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13}\\ \sigma_{21} &\sigma_{22} & \sigma_{23}\\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} \begin{Bmatrix} n_1\\ n_2\\ n_3 \end{Bmatrix}$$

$$t_i=\sum_{j=1}^{3}{\sigma_{ij}n_j}$$ $$i=1,2,3$$

$$\left\{ti \right\}_{3x1}=\left[\sigma_{ij} \right]_{3x3}\left\{n_j \right\}_{3x1}$$