The ‘σ’ of a material is 0.20. If a longitudinal strain of 4.0 x 10⁻³ is caused, by what percentage will the volume change

The correct answer is 0.24 %.

Step-by-Step Derivation and Calculation:

Poisson's ratio ($\sigma$) is defined as the ratio of lateral strain to longitudinal strain:
$\sigma =-\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}$
This gives the lateral strain as:
$\text{Lateral Strain}=-\sigma \times \text{Longitudinal Strain}$

For a cylindrical body of length L and radius r, the volume V is:
$V=\pi r^{2}L$

Taking the fractional change in volume (volumetric strain), we get:
$\frac{dV}{V}=2\frac{dr}{r}+\frac{dL}{L}$
Here, $\frac{dr}{r}$ represents the lateral strain, and $\frac{dL}{L}$ represents the longitudinal strain (${\epsilon }_L$).

Substituting the expression for lateral strain:
$\frac{dV}{V}=2(-\sigma \times {\epsilon }_L)+{\epsilon }_L$
$\frac{dV}{V}={\epsilon }_L(1-2\sigma )$

We are given the following values:
Poisson's ratio, $\sigma =0.20$
Longitudinal strain, ${\epsilon }_L=4.0\times {10}^{-3}$

Now, substitute these values into the volumetric strain formula:
$\frac{dV}{V}=(4.0\times {10}^{-3})\times (1-2\times 0.20)$
$\frac{dV}{V}=(4.0\times {10}^{-3})\times (1-0.40)$

$\frac{dV}{V}=2.4\times {10}^{-3}$

To find the percentage change in volume, multiply the fractional change by 100:
Percentage change in volume $=\frac{dV}{V}\times 100$
Percentage change in volume