Problem $5\mathrm{.}23$

Problem Statement: A block of isotropic material is stressed in the $x-$ and $y-$directions as shown

in Fig. P$5\mathrm{.}23.$ The ratio of the magnitudes of the two stresses is a constant, so that ${}_{yy}={}_{}.$

a$)$ Determine the stiffness in the $x-$direction, $E^{\text{'}}=\frac{{}_{}}{{}_{}},$ as a function of only $$ and the

elastic constants $E$ and $v$ of the material.

b$)$ Compare this apparent modulus $E^{\text{'}}$ with the elastic constant $E$ as obtained from a uniaxial

test, and comment on the comparison. $($Suggestion: Assume that $v=0\mathrm{.}3$ and consider $$

values of $-1,0,$ and $+1.)$