A rectangular block is subjected to a direct longitudinal compressive stress ${}_1.$ External pressures of intensity ${}_2$ and ${}_3$ are applied in the lateral directions perpendicular to the longitudinal direction as shown in Figure $3\mathrm{.}17,$ so that the net lateral strains in the directions along ${}_2$ and ${}_3$ are reduced to one$-$half and one$-$third, respectively, of the lateral strains that would have occurred if ${}_1$ alone was acting. Determine: $($a$){}_2$ and ${}_3$ in terms of ${}_1$ and the strain in its own direction, $($b$)$ modified value of modulus of elasticity, if the modulus of elasticity and Poisson's ratio for the material of the block are $200$ GPa and $v=0\mathrm{.}25,$ respectively, and $($c$)$ show that a material with Possion's ratio $0\mathrm{.}5$ is practically incompressible.

Solution:

$($a$)$ Lateral strain without lateral pressure $=\frac{v{}_1}{E}$

Lateral strain with lateral pressure,

${}_2=\frac{1}{E}[-{}_2+v({}_1+{}_3)]=\frac{1}{2}\frac{v{}_1}{E}$