# Introduction to Elasticity/Plane strain

A displacement field ${\displaystyle \textstyle \mathbf {u} (\mathbf {X} )}$ is called plane if

${\displaystyle u_{\alpha }=u_{\alpha }(X_{1},X_{2})\,}$

and

${\displaystyle u_{3}=0\,}$

The corresponding strain field is called a plane strain and satisfies

${\displaystyle \varepsilon _{\alpha \beta }={\cfrac {1}{2}}\left({\cfrac {\partial {u_{\alpha }}}{\partial X_{\beta }}}+{\cfrac {\partial {u_{\beta }}}{\partial X_{\alpha }}}\right)}$

and

${\displaystyle \varepsilon _{13}=\varepsilon _{23}=\varepsilon _{33}=0}$

This happens where one dimension is very large compared to the others, the principal strain in the direction of the longest dimension is constrained and can be assumed as zero, yielding a plane strain condition.