# 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 the 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 the 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)

Where,

μ = Poisson's Ratio

εₗₐₜ = Lateral Strain

εₗₒₙ₉ = Longitudinal Strain

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 the 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

## FAQ Related to Poisson's Ratio

### What is meant by Poisson's ratio?

Poisson's ratio is a measure of the ratio of a material's transverse strain (or change in width) to its longitudinal strain (or change in length) when it is stretched or compressed.

### Why is Poisson's ratio formula negative?

Poisson's ratio formula is typically defined as the negative ratio of the transverse strain to the longitudinal strain. This negative sign reflects the fact that the material is contracting in the transverse direction as it is being stretched in the longitudinal direction. Here, Compressive deformation is considered negative.

### Can Poisson's ratio be greater than 1?

No, Poisson's ratio can not be greater than 1 because it is defined as the ratio of transverse strain to longitudinal strain and these are both positive numbers, so it is always between 0 and 0.5 for most materials.

### What does the Poisson ratio of 0.5 means?

A Poisson's ratio of 0.5 means that for every unit of longitudinal strain, there is an equal amount of transverse strain in the material. In other words, the material is expanding or contracting equally in all directions when it is stretched or compressed.
This is an ideal case for isotropic materials, which have the same mechanical properties in all directions.

### What if the Poisson ratio is zero?

A Poisson's ratio of zero would indicate that there is no transverse or longitudinal strain in the material when it is stretched or compressed. This means that the material is not expanding or contracting.
This is an ideal case, and it is not possible for any real-world material to have a Poisson ratio of exactly 0. However, a material with a very low Poisson ratio, approaching 0, can be considered an approximately incompressible material.

### Why is Poisson's ratio always positive?

Poisson's ratio is always positive because it is defined as the ratio of transverse strain to longitudinal strain when a material is stretched or compressed.
Both transverse and longitudinal strains are positive numbers, meaning that they are always greater than zero. Therefore, the ratio of these two positive numbers will always be a positive number as well.

### What is the Poisson ratio of steel?

The Poisson ratio for steel is typically around 0.3. This value can vary depending on the specific type of steel and the conditions under which it is tested.

### What is Poisson's ratio formula?

Poisson's ratio formula is defined as the ratio of the transverse strain to the longitudinal strain of a material when it is stretched or compressed. Mathematically, it can be represented as:
μ = - εₗₐₜ/εₗₒₙ₉
Where,
μ = Poisson's Ratio
εₗₐₜ = Lateral Strain
εₗₒₙ₉ = Longitudinal Strain

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

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