Plane strain

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

$u_\alpha = u_\alpha(X_1, X_2) \,$

and

$u_3 = 0 \,$

The corresponding strain field is called a plane strain and satisfies

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

and

$\varepsilon_{13}= \varepsilon_{23}= \varepsilon_{33}=0$

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