Problem $2$

A sample is constrained by a rigid die and subjected to a compressive stress $({}_{zz})$

as shown:

$x$

a$)$ What are the stresses in the $x-$and $y-$directions?

b$)$ Determine the effective modulus, $E_{\text{e}\text{f}\text{f}}=\frac{{}_{zz}}{{}_{zz}}$ in the direction of the of

applied stress in terms of the elastic constants $E$ and $v$ for the material.

How is the effective modulus different from the elastic modulus obtained

from uniaxial loading?

c$)$ What happens if Poisson's ratio for the material approaches $0\mathrm{.}5?$