A deformation field is in plane strain when strain components involving one direction are negligible. Taking $z$ as that direction,

$$\varepsilon_z\approx0,\qquad \gamma_{xz}\approx0,\qquad \gamma_{yz}\approx0.$$

The body can still carry a nonzero through-thickness stress $\sigma_z$. That stress is often required to suppress the Poisson deformation that the material would otherwise undergo.

When the approximation is useful

Plane strain can approximate a long body whose geometry and loading change little along its length, provided attention is focused far from free ends. It is also important near the mid-thickness of sufficiently thick cracked specimens, where surrounding material strongly constrains out-of-plane deformation.

Plane strain versus plane stress

The two approximations remove different quantities:

• plane stress: out-of-plane stresses are negligible, but out-of-plane strain may occur;
• plane strain: out-of-plane strains are negligible, but out-of-plane stress may occur.

This distinction matters because constraint changes the local mechanical response. In fracture mechanics, thick specimens can approach plane strain and exhibit greater crack-tip constraint than thin plane-stress specimens.