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

Answer: 2

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
$=\frac{F}{Y}+\left(\frac{-2\sigma F}{Y}\right)$
$=(1-2\sigma )\frac{F}{Y}$
$\Rightarrow x=2$