Question

There is no change in the volume of a wire due to a change in its length on stretching. What is Poisson's ratio of the material of the wire?

Answer 1

Step 1: Definition of Poisson ratio

1. Poisson's ratio is the ratio of lateral strain to longitudinal strain in the direction of the stretching force.
2. It is a scalar quantity.
3. By sign convention, tension deformation is considered positive and compression is considered negative. For most materials, the value of Poisson’s ratio lies in the range, $0$ to $0.5$.
4. The Poisson's ratio includes a negative sign, so ordinary materials have positive numbers. Poisson's ratio, also known as Poisson's coefficient, is usually represented by $\mu$.

Mathematically, it can be expressed as,

$\mu =-\frac{\mathrm{Lateral}\mathrm{strain}}{\mathrm{Longitudinal}\mathrm{strain}}$

Step 2: Calculate the Poisson's ratio

The volumetric strain in terms of longitudinal strain and Poisson's ratio is defined as the product of longitudinal strain and the term$(1–2\mu )$, where $\mu$ is Poisson's ratio and is given as:

$\frac{dV}{V}=(1–2\mu )\frac{dL}{L}$

Where, $dV$is the change in volume, $V$ is the initial volume, $dL$ is the change in length, $L$ is the initial length, $\frac{dV}{V}$is volumetric strain, $\mu$ is Poisson's ratio of the material of the wire, $\frac{dL}{L}$ is longitudinal strain.

As there is no change in the volume of wire, hence $\frac{dV}{V}=0$.

then,

$(1–2\mu )\frac{dL}{L}=0$

$1–2\mu =0$

$\mu =0.5$

The value of Poisson's ratio of the material of the wire is $0.5$.