What is Poisson’s Ratio? Definition, Meaning, Formula, Examples, Equation

Let’s explore Poisson’s ratio!

What is Poisson’s Ratio? Definition, Meaning, Formula, Examples, Equation

What is Poisson’s Ratio along with the definition, meaning, formula, equation, examples, list, exercise, etc. are captured to have a clear concept.

In addition, we will see here the value of Poisson’s ratio of steel, aluminium, concrete, wood, etc. most widely used materials.

Let’s explore this Ratio!

What is Poisson’s Ratio?

Let’s understand the basics of Poisson’s ratio along with explanation, meaning & example.

Poisson’s Ratio Basics

In any industry, we use a lot of pipes with different materials, different kinds of beams, and other many various materials that need to be tested by stretching.

For example, the material needs to be tested its strength by stretching it.

Now, take a bar and stretch it. What are the things you observe?

  • Force is applied along the length, as it will be stretching along the length.
  • Its length will increase or elongation happens due to force. We can say, Longitudinal or Axial Strain
  • Compression will happen perpendicular to the force application that is across the width or transverse direction of the bar, or we can say, lateral or transverse strain happens.
what is poissons ratio meaning definition formula equation basics
Fig. 1 What is Poisson’s ratio meaning definition formula equation basics

Now, this elongation (Longitudinal or Axial Strain)  or compression (lateral or transverse strain) is different in different materials. Now, how much strain is created or what is the relationship between these strains is necessary to describe for each material and the concept of Poisson’s Ratio comes into the picture.

Poisson’s Ratio Example & Explanation

Take a rubber and try to stretch along its length. You will see it is easy to stretch and length is increased and simultaneously its width is reduced.

Why does it happen? Because

  • Force is applied on the rubber strip along the length.
  • The rubber strip length will increase and it will be in the longitudinal direction and called as Longitudinal strain or Axial Strain.
  • The thickness of the rubber is reduced i.e. lateral strain or transverse strain.

Now, Poisson’s ratio relates to the relationship between these two strains.

Poisson Ratio Definition & Poisson’s Effect

The Poisson’s ratio is one of the main factors which is defined as the ratio of the transverse or lateral strain to that of the longitudinal or axial strain in the direction of the stretching force.

This phenomenon is called Poisson’s effect.

  • It is a basic material property of materials.
  • It is a constant value specific to materials.
  • This value is considered within the elastic limit.
  • Different materials have different Poisson ratio.
  • Greek letter ν (nu) is used to denote this ratio.
  • It is negative for isometric materials.
  • It is positive for anisometric materials.

We will discuss mainly isometric materials here.

Hence, as per definition,

Poisson’s Ratio = (-) Transverse or lateral strain / Longitudinal or Axial strain

[Negative symbol implies that during elongation there will be a decrease of width/diameter]

Where,

  • Lateral or Transverse strain = change in diameter or width/original diameter or width
  • Longitudinal or Axial Strain = change in length/original length

What does Poisson ratio of 0.5 mean?

Poisson’s ratio for a material 0.5 means a factor that implies that deformation happens due to the application of force on the material. Lateral deformation or lateral strain is 0.5 times the longitudinal deformation or longitudinal strain of that material.

History

Siméon Poisson has developed the concept and it is named Poisson ratio based on his name.

Poisson’s Ratio Formula & Equation

Take a cylindrical rubber piece to deduce the formula or equation of the Poisson ratio.

Consideration

A force is applied on the rubber piece along the length to stretch and its original length and diameter are changed.

Consider,

  • F = Force application along the length
  • lo = Original length of the rubber piece
  • do = Original diameter of the rubber piece
  • l = Final length after the application of force
  • d = Final diameter after the application of force
  • Δl = Change in length (l-lo)
  • Δd = Change in diameter (d-do)

Diagram

Let’s see the diagram to understand Poisson’s ratio formula & equation,

poissons ratio formula equation
Fig. 2 Poisson’s ratio formula equation

Poisson’s Ratio Formula or Equation Deduction

Based on the definition of Poisson Ratio,

We can write,

Poisson Ratio, σ = (-) Lateral or Transverse Strain/Longitudinal or Axial Strain

Lateral or Transverse strain

= change in diameter/original diameter

= Δd/ d

= (d-do)/ d

Longitudinal or Axial Strain

= change in length/original length

= Δl/ l

= (l-lo)/ l

Hence, from the definition of Poisson’s Ratio, we write,

Poisson’s Ratio,

σ = (-)[(d-do)/ d]/[ (l-lo)/ l]

σ = – l(d-do)/d(l-lo)

This is the simple derivation of Poisson’s Ratio Formula or Equation.

Properties of Poisson’ Ratio

Unit of Poisson’s Ratio

Poisson Ratio, σ = Lateral or Transverse Strain/Longitudinal or Axial Strain

Now,

The unit of Poisson’s Ratio

= Unit of Lateral or Transverse Strain/Unit of Longitudinal or Axial Strain

Now,

Strain = change in dimension/original dimension.

  • Unit of strain = unit of change in dimension/unit of the original dimension
  • Unit of strain = unit of dimension/unit of dimension = unitless, hence,

Transverse strain = unit less

Longitudinal strain = unit less

Hence, The unit of Poisson’s Ratio

= unit less / unit less

= unit less

Is it Scalar or Vector?

As Poisson’s ratio doesn’t specify any direction, it is simply a scalar quantity.

Range

Normally, Poisson’s ratio lies in the range of -1.0 to +0.5. However, in case of anisometric materials, it can be more than 0.5 as well.

Poisson Ratio relation with Temperature

Let’s see the effect of temperature on Poisson Ratio. Consider two conditions,

  • Hot condition
  • Cold condition

In the case of the hot condition or cold condition, both lateral strain and longitudinal strain are supposed to change simultaneously. But due to the behavior of the material with respect to the temperature, it is observed that this ratio slightly decreases with temperature.

Negative or Positive

Tensile deformation means an increase in length by the original length. Here, an increase means positive.

Compressive deformation means a decrease in width by original width. Here, decrease means negative.

Hence, Poisson’s Ratio always represents = negative/positive = negative. [ -ve means here decrease in diameter during elongation]

This is applicable to all isotopic materials.

But what about the anisotropic materials? It is different. In case of anisotropic materials, the inter-atomic structure is re-arranged under tensile stress and we get a positive transverse strain. Hence, Poisson’s Ratio will be positive for these cases.

Poisson’s ratio can also be greater than 1

We have seen Poisson’s ratio maximum of 0.5. This value is for isotropic materials only. However, this value can be more than 1 as well for anisotropic materials like polyurethane foam.

Maximum Poisson’s ratio

The maximum Poisson’s ratio is for rubber and it is about 0.4999. This is for only isotropic materials only.

Poisson’s Ratio of Steel, Concrete, Aluminium, Rubber & Others

Poisson’s ratio of steel (0.30)

Poisson’s ratio for steel is 0.3 and it implies that deformation happens due to the application of force on the steel. Lateral deformation or lateral strain is 0.3 times of the longitudinal deformation or longitudinal strain of steel.

Poisson’s ratio aluminium (0.32)

Poisson’s ratio of aluminium is 0.32 and it implies that deformation happens due to the application of force on the aluminium.

Lateral deformation or lateral strain is 0.32 times of the longitudinal deformation or longitudinal strain of aluminium.

Poisson’s ratio for Concrete (0.1 to 0.2)

Poisson’s ratio of concrete varies from 0.1 to 0.2 and it implies that deformation happens due to the application of force on the concrete.

Lateral deformation or lateral strain is in the range of 0.1 to 0.2 times of the longitudinal deformation or longitudinal strain of concrete.

Poisson’s ratio of wood (0.2 to 0.4)

Poisson’s ratio of wood varies from 0.2 to 0.4 and it implies that deformation happens due to the application of force on the wood.

Lateral deformation or lateral strain is in the range of 0.2 to 0.4 times of the longitudinal deformation or longitudinal strain of wood.

Poisson’s ratio of rubber (0.4999)

Poisson’s ratio of rubber is 0.4999 and it implies that deformation happens due to the application of force on the rubber.

Lateral deformation or lateral strain is 0.4999 times of the longitudinal deformation or longitudinal strain of rubber.

Poisson’s Ratio of Cork (0.0)

The poisson ratio of cork is approximately 0.0 as it doesn’t exhibit any transverse strain during its stretching.

Other materials

Poisson ratio other few materials are tabulated for reference,

Material Poisson Ratio
Brass 0.331
Bronze 0.34
Carbon Steel 0.27–0.30
Cast iron 0.21–0.26
Clay 0.30–0.45
Concrete 0.1–0.2
Copper 0.33
Cork 0.0
Foam 0.10–0.50
Glass 0.18–0.3
Gold 0.42–0.44
Granite 0.2 – 0.3
Inconel 0.27-0.38
Ice 0.33
Lead 0.431
Limestone 0.2-0.3
Magnesium alloy 0.252–0.289
Marble 0.2 – 0.3
Molybdenum 0.307
Monel 0.315
Nickel 0.31
Nickel Steel 0.291
Platinum 0.39
Polystyrene 0.34
Rubber 0.48-0.4999
Sand 0.20–0.455
Silver 0.37
Stainless steel 0.30–0.31
Tin 0.33
Titanium 0.265–0.34
Tungsten 0.28
Wrought iron 0.278
Zinc 0.331

How to Measure Poisson’s Ratio?

We have already seen the values of the Poisson ratio for different materials. Now, how to measure this value? Let’s try to understand,

  • Need a precision machine to measure the deformation, extensometer is widely used in this regard.
  • Take a specimen for which the ratio to be measured.
  • Measure the length.
  • Measure the width or diameter of the specimen.
  • Mount it properly in the extensometer.
  • Apply force for the extension.
  • Measure the length & width after extension.

Now, calculate the transverse & longitudinal strain and divide it as per definition to get the value.

Application of Poisson’s Ratio

The application of this ratio is huge in many industries, a few of them are listed below,

Piping & Pipeline industries

Piping and pipelines are very common in many industries like power plants,  water projects, oil & gas projects, HVAC systems, etc. and material selection is very crucial. This ratio helps to select proper materials based on the applications.

Automobile Industries

In the automobile industry, various types of metals are used and the same is selected based on the strength as well as this ratio.

Aviation Industries

There are high-strength and lightweight materials used in aviation industries and the Poisson ratio is used to select the materials.

Cork used for bottle sealing

Cork is considered as 0, and in case of cork, there is no lateral strain under axial strain. This material is used for perfect bottle sealing.

Material Strength

This ratio provides the data of material strength which helps to select the materials.

Exercise on Poisson’s Ratio

Question

When a metallic rod of 5mm diameter is subjected to a tensile force of 5 x 103 N, there will be a change in diameter by 3.5 x 10-4 cm.

Calculate the Poisson ratio. If the Young modulus of the metallic rod is 8 x 1010 N/m2.

Given Data

Metallic rod diameter = d = 5 mm = 5 x 10-3 m

Area of rod, A = πd2 = 3.14 x (5 x 10-3)2

Applied force, F = 5 x 103 N

Diameter change due to force = Δd = 3.5 x 10-4 cm = 3.5 x 10-6 m

Young modulus of the metal, Y = 8 x 1010 N/m2

Calculate

Calculate the value of Poisson Ratio.

Solution

Poisson ratio (ν) = (-)Lateral strain / Longitudinal strain

Lateral strain

= Δd /d

= 3.5 x 10-6 / 5 x 10-3

= 3.5 x 10-3/5

= 7 x 10-4

Now, to calculate the Poisson ratio, we have to get the value of longitudinal strain as well, So, how do we get this value?

Here, we have to think about the young modulus!

We all know that,

Young modulus, Y = Longitudinal stress / Longitudinal strain

Longitudinal stress

= Force/area

= F / A

= 5 x 103 / 3.14 x (5 x 10-3)2

So,

Y = Longitudinal stress / Longitudinal strain

8 x 1010 = [5 x 103 / 3.14 x (5 x 10-3)2] / Longitudinal strain

Longitudinal strain = 5 x 103 / 3.14 x (5 x 10-3)2/8 x 1010

Longitudinal strain

= 5 x 103 / 3.14 x (5 x 10-3)2 x 8 x 1010

= 5 x 103 / 3.14 x 5 x 8 x 104

= 1 / 3.14 x 8 x 10

= 0.003981

= 3.981 x 10-3 m

So, Poisson Ratio, ν

=7 x 10-4 / 3.981 x 10-3

= 1.7583 x 10-1

= 0.175 (result)

So, the value of the ratio is 0.175

Code & Standards for Poisson Ratio

To measure this ratio, the below standard is used.

  • ASTM E132 Standards for Poisson ratio measurements

Conclusion

Hence, we have got a basic idea about Poisson Ratio along with the definition, meaning, formula, examples, a small exercise, etc. Any doubt, please let us know. Happy reading!

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