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 $($see figure below$).$ The strains associated with length, i$.$e$.,$

the normal strain ${}_{zz}$ and the shear strains ${}_{xz}$ and ${}_{yz}($if the length is the z$-$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:

$=\left[\begin{array}{ccc}{}_{}& {}_{xy}& 0\\ {}_{yx}& {}_{yy}& 0\\ 0& 0& 0\end{array}\right]$

For isotropic material at plane strain state, demonstrate that the stress along the length

direction $(z-$direction$)$ is given by

${}_{zz}=v({}_{}+{}_{yy})=({}_{}+{}_{yy})$