User:Eas4200c.f08.WIKI.E/hw4

 Friday, October 17 

Normal strains

$$\varepsilon _{xx}=\frac{\sigma _{xx}}{E}-\frac{\nu \sigma _{yy}}{E}-\frac{\nu\sigma _{zz} }{E} $$

$$\varepsilon _{yy}=-\frac{\nu\sigma _{xx}}{E}+\frac{ \sigma _{yy}}{E}-\frac{\nu\sigma _{zz} }{E} $$

$$\varepsilon _{zz}=-\frac{\nu\sigma _{xx}}{E}-\frac{\nu \sigma _{yy}}{E}+\frac{\sigma _{zz} }{E} $$

Shear Strains

$$\gamma _{xy} =2 \zeta _{xy}= \frac{\tau_{xy}}{G}=\frac{\sigma_{xy}}{G}$$

$$\gamma _{yz} =2 \zeta _{yz}= \frac{\tau_{yz}}{G}=\frac{\sigma_{yz}}{G}$$

$$\gamma _{zx} =2 \zeta _{zx}= \frac{\tau_{zx}}{G}=\frac{\sigma_{zx}}{G}$$

Unroll this matrix along the diagonal and then up and around counterclockwise.

$$\epsilon = \begin{vmatrix} \varepsilon_ {11} & \varepsilon_{12} & \varepsilon _{13}\\ \varepsilon _{21} &\varepsilon _{22} &\varepsilon _{23} \\ \varepsilon _{31} &\varepsilon _{32} &\varepsilon _{33} \end{vmatrix}$$

Due to the symmetry, you may rewrite this unrolled matrix into a $$6_x 1$$ with a Voigt Notation:

$$ \left\{\varepsilon _{ij} \right\}_{6x1}= \begin{Bmatrix} \varepsilon _{11}\\ \varepsilon _{22}\\ \varepsilon _{33}\\ \varepsilon _{23}\\ \varepsilon _{13}\\ \varepsilon _{12} \end{Bmatrix}$$

$$\left\{\sigma _{ij} \right\}_{6x1}= \begin{Bmatrix} \sigma _{11}\\ \sigma _{22}\\ \sigma _{33}\\ \sigma _{23}\\ \sigma _{13}\\ \sigma _{12} \end{Bmatrix}$$

Hooke's Law for isotropic elasticity:

$$\begin{Bmatrix} \varepsilon _{11}\\ \varepsilon _{22}\\ \varepsilon _{33}\\ \gamma _{23}\\ \gamma _{13}\\ \gamma _{12} \end{Bmatrix}=

\begin{vmatrix} \frac{1}{E}& \frac{-\nu }{E} & \frac{-\nu }{E} & 0 &0 &0 \\ \frac{-\nu }{E}& \frac{1}{E} &\frac{-\nu }{E}  & 0 & 0 & 0\\ \frac{-\nu }{E} & \frac{-\nu }{E} & \frac{1}{E} & 0 &0  &0 \\ 0& 0 & 0 & \frac{1}{G} & 0 & 0\\ 0& 0 & 0 & 0 & \frac{1}{G} &0 \\ 0& 0 & 0 & 0 & 0 & \frac{1}{G} \end{vmatrix} \begin{Bmatrix} \sigma _{11}\\ \sigma _{22}\\ \sigma _{33}\\ \sigma _{23}\\ \sigma _{13}\\ \sigma _{12} \end{Bmatrix}$$

It can also be written as

$$\begin{Bmatrix} \varepsilon _{11}\\ \varepsilon _{22}\\ \varepsilon _{33}\\ \varepsilon _{23}\\ \varepsilon _{13}\\ \varepsilon _{12} \end{Bmatrix}=

\begin{vmatrix} \frac{1}{E}& \frac{-\nu }{E} & \frac{-\nu }{E} & 0 &0 &0 \\ \frac{-\nu }{E}& \frac{1}{E} &\frac{-\nu }{E}  & 0 & 0 & 0\\ \frac{-\nu }{E} & \frac{-\nu }{E} & \frac{1}{E} & 0 &0  &0 \\ 0& 0 & 0 & \frac{1}{2G} & 0 & 0\\ 0& 0 & 0 & 0 & \frac{1}{2G} &0 \\ 0& 0 & 0 & 0 & 0 & \frac{1}{2G} \end{vmatrix} \begin{Bmatrix} \sigma _{11}\\ \sigma _{22}\\ \sigma _{33}\\ \sigma _{23}\\ \sigma _{13}\\ \sigma _{12} \end{Bmatrix}$$

Poisson's ratio values for different materials
Poisson's Ratio table []