# Poisson's Ratio: Definition, Formula, Unit, Calculation

In this article, you will learn a complete overview of Poisson's ratio such as its definition, formula, unit, calculation, and many more.

Poisson's ratio is one of the types of elastic constant.

Elastic constants are those constants that determine the deformation produced by a given stress system acting on any material.

The elastic constant measures the elasticity of the material.

It is used to derive the relationship between stress and strain.

For a homogeneous and isotropic material, there are four types of elastic constants:
1. Modulus of Elasticity  or Young’s modulus
2. Modulus of rigidity or Shear modulus
3. Bulk modulus
4. Poisson’s Ratio

Here we only discussed Poisson's ratio.

Modulus of elasticity, modulus of rigidity, and bulk modulus are already discussed in our previous article.

## Definition of Poisson's Ratio

When a material is loaded within elastic limit then, the ratio of lateral strain to linear strain remains constant and is called  Poisson's ratio

Poisson's ratio is denoted by μ or 1/m.

Its value ranges from 0.1 to 0.5.

Mathematically,

Lateral Strain ∝ Longitudinal Strain

εₗₐₜ ∝  εₗₒₙ₉

So,

μ = Longitudinal Strain/ Lateral Strain

μ   =  - εₗₐₜ/εₗₒₙ

Where,

εₗₐₜ = lateral strain

εₗₒₙ = longitudinal strain

μ = Poisson's ratio

### Longitudinal Strain

When an elastic body is subjected to external deforming force then there is a change in the dimension of the body, the dimension along the direction of force is called as longitudinal dimension and the ratio of change in the longitudinal dimension to the original dimension is called as longitudinal strain.

### Lateral Strain

When an elastic body is subjected to external deforming force then there is a change in the dimension of the body, the dimension perpendicular to the direction of force is called lateral dimension, and the ratio of change in lateral dimension to original dimension is called lateral strain.

## Poisson's Ratio Formula

Let the length of a cylinder be l and the diameter d.

Whose bottom part is fixed, now this cylinder is pulled out by applying an external force, due to which its length increases to l' and diameter decreases to d'.

So,

Longitudinal change in length,

Δl = l' - l

Lateral change in length,

Δd = d' - d

Clearly shown in figure d greater than d' so Δd will be negative.

As we know,

Strain = Change in length/Original Length

So,

εₗₒₙ₉ = Δl/l

εₗₐₜ = Δd/d

So,

Poisson's  ratio

μ = - εₗₐₜ/εₗₒₙ₉

μ = - (Δd/d)/(Δl/l)

A negative sign is shown here because when longitudinal strain increases then lateral strain decreases so that the value of εₗₐₜ will be negative and when the lateral strain increases then longitudinal strain decreases so that the value of εₗₒₙ₉ will be negative so it always gives a negative sign.

So Poisson's ratio for a material is the negative of the ratio of lateral strain to the longitudinal strain of a body made of that material.

## Unit of Poisson's Ratio

As we know strain is a unit less quantity, and Poisson's ratio is the ratio of lateral strain to linear strain, so it will also be a unit less.

## Poisson's Ratio Value For Different Materials

Material Poisson's Ratio
Cork 0
Concrete 0.1 - 0.2
Aluminum 0.33
Cast iron 0.2 - 0.3
Steel 0.27 - 0.3
Elastic Rubber 0.5
Gold 0.42 - 0.44
Glass 0.18 - 0.3
Copper 0.33
Clay 0.30 - 0.45
Foam 0.10 - 0.50

So, the lower the value of Poisson's ratio, the more resistant the material is to deformation.

For example, rubber has a Poisson ratio of 0.5 which can be easily stretched, whereas cork has a Poisson ratio of zero so that even when compressed or stretched, the width or diameter of the material remains the same.

## Calculation of Poisson's Ratio

### Question

In an experiment, a bar of 30 mm diameter is subjected to a pull of 80 KN. The measured extension of gauge length of 200 mm is 0.09 mm and the change in diameter is 0.0039 mm. Calculate the Poisson's ratio.

### Solution

Given Data,

P = 80 KN

L = 200 mm

d = 30 mm

Δl = 0.09 mm

Δd = 0.0039 mm

μ =?

As we know,

Lateral strain,

εₗₐₜ = Δd/d

εₗₐₜ = 0.0039/30

εₗₐₜ = 1.3 × 10⁻⁴

Longitudinal Strain,

εₗₒₙ₉ = Δl/l

εₗₒₙ₉ = 0.09/200

εₗₒₙ₉ = 4.5 × 10⁻⁴

So,

Poisson's Ratio,

μ = - εₗₐₜ/εₗₒₙ₉

μ = - (1.3 × 10⁻⁴)/(4.5 × 10⁻⁴)

μ = 0.289

So here you have to know all aspects related to Poisson's ratio. If you have any doubt then you are free to ask me by mail or on the contact us page.

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