where $E$ and

$u$ are the Young's modulus and Poisson's ratio.

For a material that behaves according to the generalized Hooke's law:

$($a$)$ Considering the case of plane stress $(xy-$plane$),$ derive the strain transformation

equations from the stress transformation equations.

$($b$)$ How does the strain component ${}_{zz}$ transform in part $($a$)?$

$($c$)$ Considering the case of plane strain $(xy-$plane$),$ derive the stress transformation

equations from the strain transformation equations.

$($d$)$ How does the stress component ${}_{zz}$ transform in part $($c$)?$

$*$Note that the strains have been written in terms of the symbol for micron, $,$ i$.$e$.1={10}^{-6}.$