A material has Poisson 's ratio 0 50 If a unform rod of it suffers a longitudinal strain of - Physics - Mechanical Properties Of Solids

Question

A material has Poisson 's ratio 0 50 If a unform rod of it
suffers a longitudinal strain of 2x10^-3 Percentage change in
volume is

Neha Sarin · Accepted Answer

Formula for Poisson's ratio is

$nu = -frac{dvarepsilon_mathrm{trans}}{dvarepsilon_mathrm{axial}} = -frac{dvarepsilon_mathrm{y}}{dvarepsilon_mathrm{x}}= -frac{dvarepsilon_mathrm{z}}{dvarepsilon_mathrm{x}}$

where

$nu$ is the resulting Poisson's ratio,

$varepsilon_mathrm{trans}$ is transverse strain (negative for axial tension (stretching), positive for axial compression)

$varepsilon_mathrm{axial}$ is axial strain (positive for axial tension, negative for axial compression).

And the volumetric change is given by

Using $V=L^3$ and $V+Delta V=(L+Delta L)(L-Delta L')^2$ :

$frac {Delta V} {V} = left(1+frac{Delta L}{L} right)left(1-frac{Delta L'}{L} right)^2-1$

Using the above derived relationship between $Delta L$ and $Delta L'$ :

$frac {Delta V} {V} = left(1+frac{Delta L}{L} right)^{1-2nu}-1$

and for very small values of $Delta L$ and $Delta L'$ , the first-order approximation yields:

$frac {Delta V} {V} approx (1-2nu)frac{Delta L}{L}$

Using the above formula you can calculate your answer

A material has Poisson 's ratio 0.50. If a unform rod of it suffers a longitudinal strain of 2x10^-3. Percentage change in volume is