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# Plane strain

In real engineering components, stress (and strain) are 3-D tensors but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. The strains associated with length, i.e the normal strain ε33 and the shear strains ε13 and ε23 (if the length is the 3-direction) are constrained by nearby material and are small compared to the cross-sectional strains. The strain tensor can then be approximated by: $\underline{\underline{\epsilon}} = \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & 0 \\ \epsilon_{21} & \epsilon_{22} & 0 \\ 0 & 0 & 0\end{bmatrix}$

### Additional recommended knowledge

in which the double underline indicates a second order tensor. This strain state is called plane strain. The corresponding stress tensor is: $\underline{\underline{\sigma}} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & 0 \\ \sigma_{21} & \sigma_{22} & 0 \\ 0 & 0 & \sigma_{33}\end{bmatrix}$

in which the non-zero σ33 is needed to maintain the constraint ε33 = 0. This stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3-D problem to a much simpler 2-D problem.