User:EAS4200C.F08.WIKI.A/My Contribution: 6.1



Here we will find the sum of the forces in the y-direction.
$$\displaystyle \begin{Vmatrix} \bar{n} \end{Vmatrix}= 1$$ $$\displaystyle \sum{F_y} = -\sigma_{yy}\cdot (dz\cdot 1) - \sigma(dy\cdot1) + t_y(ds\cdot1) = 0 = \begin{cases}dz =ds\cos \theta \\ dy = ds\sin\theta \end{cases}$$

$$\displaystyle 0 = -\sigma_{yy}n_y - \sigma_yzn_z + t_y ds$$

$$\displaystyle n_y = \cos\theta, n_z = \sin\theta $$ is the unit normal vector unit depth along x-direction.

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:

Taking the sum of the forces in the z-direction
$$t_z=\sigma _{yz}\cdot n_y+\sigma _{zz}n_z$$

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

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

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

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