6.3 Plane stress
When a body is predominately loaded in two directions and unloaded in the third, a plane stress assumption can be made.
Plane stress
- Recall: \[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{zz}}\cr {\varepsilon_{yz}}\cr {\varepsilon_{xz}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E} & \frac{-\nu}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \cr \frac{-\nu}{E} & \frac{1}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \cr \frac{-\nu}{E} & \frac{-\nu}{E} & \frac{1}{E} & 0 & 0 & 0 \cr 0 & 0 & 0 & \frac{1}{2 G} & 0 & 0 \cr 0 & 0 & 0 & 0 & \frac{1}{2 G} & 0 \cr 0 & 0 & 0 & 0 & 0 & \frac{1}{2 G} \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr 0 \cr 0 \cr 0 \cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\]
The result is a simplified set of governing constitutive equations: \[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{ccc} \frac{1}{E} & \frac{-\nu}{E} & 0 \cr \frac{-\nu}{E} & \frac{1}{E} & 0 \cr 0 & 0 & \frac{1}{2 G} \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\]
\[\begin{equation*} \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{ccc} \frac{E}{1-\nu^2} & \frac{\nu E}{1-\nu^2} & 0 \cr \frac{\nu E}{1-\nu^2} & \frac{E}{1-\nu^2} & 0 \cr 0 & 0 & {2 G} \cr \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} \end{equation*}\]