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.
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,
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, the extensometer is widely used in this regard.
- Take a specimen for which the ratio is 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!
Further Study
Luminous and Illuminated Objects
How does light interact with matters
