6.4 Plane strain
- When a body is predominately strained in two directions and un-strained in the third, a plane strain assumption can be made.
Plane strain
\[\begin{equation*} \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{zz}}\cr {\sigma_{yz}}\cr {\sigma_{xz}}\cr {\sigma_{xy}}\cr \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)} \left[ \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 & 0 \cr \nu & 1-\nu & \nu & 0 & 0 & 0 \cr \nu & \nu & 1-\nu & 0 & 0 & 0 \cr 0 & 0 & 0 & 1-2\nu & 0 & 0 \cr 0 & 0 & 0 & 0 & 1-2\nu & 0 \cr 0 & 0 & 0 & 0 & 0 & 1-2\nu \cr \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr 0 \cr 0 \cr 0 \cr {\varepsilon_{xy}}\cr \end{array} \right\} \end{equation*}\]
\[\begin{equation*} \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{xy}}\cr \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)} \left[ \begin{array}{cccccc} 1-\nu & \nu & 0 \cr \nu & 1-\nu & 0 \cr 0 & 0 & 1-2\nu \cr \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} \end{equation*}\]
\[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} = \frac{1+\nu}{E} \left[ \begin{array}{cccccc} 1-\nu & -\nu & 0 \cr -\nu & 1-\nu & 0 \cr 0 & 0 & 1 \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\]