Consider the figure below showing a solid cylinder $($red color$)$ made of a material

displaying isotropic linear elasticity. The cylinder fits perfectly inside a thick$-$walled

cylinder $($gray color$)$ made of an infinitely rigid $($non$-$deformable$)$ material. The elastic

cylinder $($red$)$ is in a state of uniaxial compression, as a result of the application of

stress, ${}_{zz},$ equal to $40$ MPa $.$ The Young's modulus, E$,$ of the cylinder is $100$ GPa and

Poisson's ratio is $0\mathrm{.}20.$

Determine the magnitude of $($zz$).$