Question

The stress along the length of a rod(with rectangular cross section) is $1\%$ of the Young's modulus of its material. What is the approximate percentage of change of its volume? (Poisson's ratio of the material of the rod is $0.3$).

• A. $3\%$
• B. $1\%$
• C. $0.7\%$
• D. $0.4\%$

Answer 1

The correct option is D $0.4\%$

$\Rightarrow v=xyz$
$\frac{dv}{v}=\frac{dx}{x}+\frac{dy}{y}+\frac{dz}{z}=\frac{dl}{l}-\mu \frac{dl}{l}-\mu \frac{dl}{l}=(1-2\mu )\frac{dl}{l}$
Stress $=Y\times$strain
$0.01Y=Y\frac{dl}{l}\Rightarrow \frac{dl}{l}=0.01$
$\frac{dv}{v}=0.01\times (1-0.6)=0.01\times 0.4$
$\%$ Volume change $=0.4\%$.