Question

On all the six surfaces of a unit cube, equal tensile force of $F$ is applied. The increase in length of each side is $\frac{F}{Y}-\frac{x \sigma F}{Y}$. Find value of $x$ where, $Y=$ Young's modulus, and $\sigma=$ Poisson's ratio.

Answer 1

Tensile strain on each face $=\frac{l}{ L }=\frac{ FA }{ Y }$
For $A=1$ sq. unit,
$\frac{l}{ L }=\frac{ F }{ Y }$
Lateral strain due to force acting on perpendicular face $=-\sigma \times \frac{l}{L}$
$=-\sigma \times \frac{F}{Y}$
As force is subjected from both directions,
Lateral strain $=\frac{-2 \sigma F }{ Y }$
Total increase in length is attributed to both tensile and lateral strain.
$\therefore $ Increase in length of each side
$=\frac{F}{Y}+\left(\frac{-2 \sigma F}{Y}\right)$
$=(1-2 \sigma) \frac{F}{Y}$
$\Rightarrow x=2$