A general stress state in a solid has normal and shear components acting on planes in three dimensions. Plane stress is a useful approximation when all stress components involving one direction are negligible.

For a thin body loaded mainly in its own plane, the stress state can often be represented as

$$\boldsymbol\sigma= \begin{bmatrix} \sigma_x & \tau_{xy}\ \tau_{xy} & \sigma_y \end{bmatrix}.$$

When the approximation is useful

Thin plates, thin-walled vessels and other components whose thickness is small compared with their in-plane dimensions often have very small through-thickness stress away from local load application regions.

What plane stress does not mean

It does not mean that the material is two-dimensional or that deformation out of the plane is zero. Poisson contraction can still produce through-thickness strain.

Plane stress reduces a three-dimensional stress problem to the in-plane components that dominate the mechanical response.