Plane strain

Remarks

Example: the shear $u = (\gamma x_2, 0, 0)$ is in plane strain, with $\varepsilon_{12} = \gamma/2$ as its only nonzero strain component.

Non-example: for an isotropic material with Poisson's ratio $\nu \ne 0$, uniaxial stress along $x_1$ produces the strain $\varepsilon_{33} = -\nu \varepsilon_{11} \ne 0$, so this state is not in plane strain with respect to the $x_1 x_2$-plane.

For an isotropic linear elastic material in plane strain, the out-of-plane stress is $\sigma_{33} = \nu (\sigma_{11} + \sigma_{22})$, which is not zero in general. The in-plane equations of plane strain have the same form as those of plane stress, with the elastic constants changed: $E$ is replaced by $E/(1 - \nu^2)$ and $\nu$ by $\nu/(1 - \nu)$.