Plane stress

Remarks

Example: uniaxial tension $\sigma = s \, e_1 \otimes e_1$ is in plane stress, since its only nonzero component is $\sigma_{11}$.

Non-example: the hydrostatic stress $\sigma = -p I$ with $p \ne 0$ is not in plane stress for any plane, since $\sigma e_3 = -p e_3 \ne 0$ for every unit vector $e_3$.

In the usual two-dimensional theory one also assumes that $\sigma_{11}, \sigma_{12}, \sigma_{22}$ do not depend on $x_3$. For an isotropic linear elastic material with Young's modulus $E$ and Poisson's ratio $\nu$, plane stress gives the out-of-plane strain $\varepsilon_{33} = -\frac{\nu}{E} (\sigma_{11} + \sigma_{22})$, which is not zero in general. Plane stress is exact only in special cases; for thin plates loaded in their plane it is an approximation that improves as the thickness decreases.