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 ε ε 33 and the shear strains ε ε 13 and ε ε 23 (if the length is the 3-direction) are constrained by nearby material and are small compared to the cross-sectional strains. Plane strain is then an acceptable approximation. The strain tensor for plane strain is written as: ε ε _ _ _ _ = [ ε ε 11 ε ε 12 0 ε ε 21 ε ε 22 0 0 0 0 ] in which the double underline indicates a second order tensor. This strain state is called plane strain. The corresponding stress tensor is: σ σ _ _ _ _ = [ σ σ 11 σ σ 12 0 σ σ 21 σ σ 22 0 0 0 σ σ 33 ] in which the non-zero σ σ 33 is needed to maintain the constraint ϵ ϵ 33 = 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.