Poisson’s ratio is a value that defines the amount of contraction that occurs when something is stretched, or the amount of expansion when an object is compressed. The contraction is always measured at 90^{o} to the stretching or compression force.
To put it in very simple terms – in general, when an object is stretched it tends to get thinner and when it’s compressed it gets thicker. This is known as the Poisson Effect.
Poisson’s ratio is simply a ratio of the strain in the direction of stretching, against the perpendicular strain. Strain is defined as the deformation caused by stress.
The ratio is named after Simeon Poisson, a French engineer, physicist and mathematician, who defined the Poisson effect in his 1811 book Traité de Mécanique – The Mechanics of Materials.
Poisson’s ratio can be found by measuring the differences in thickness and length during stretching or compression, then using those values to calculate the strain in each direction. The lower the value, the more resistant the material is to deformation. The values typically range between 0.1 and 0.5 for most materials.
Now that we have a very basic definition, let’s look at things in a bit more detail.
The technical definition of Poisson’s Ratio
In technical terms, Poisson’s ratio is a ratio of the transverse contraction or expansion strain (perpendicular) against the longitudinal strain (i.e. in the same direction as the applied force).
Poisson’s ratio is denoted by the Greek letter nu (μ).
Mathematically:
μ = – ε_{t} / ε_{l}
Given that:
μ = Poisson’s ratio
ε_{t }= transverse (perpendicular) strain
ε_{l }= longitudinal strain
The minus sign allows for the fact that stretching deformation is taken as positive and compressive deformation is negative.
The longitudinal strain can be calculated by using the following formula:
ε_{l} = dl / L
Given that:
ε_{l} = longitudinal strain
dl = difference in length (m)
L = static length (m)
The transverse strain can be calculated by using the following formula:
ε_{t} = dr / r
Given that:
ε_{t} = transverse strain
dr = difference in radius (m)
r = initial radius (m)
Common values of Poisson’s Ratio
This table contains values for some commonly used materials:
Material | Poisson’s Ratio |
– μ – | |
Maximum Value: | 0.5 |
Aluminum | 0.334 |
Aluminum, 6061-T6 | 0.35 |
Aluminum, 2024-T4 | 0.32 |
Beryllium Copper | 0.285 |
Brass, 70-30 | 0.331 |
Brass, cast | 0.357 |
Bronze | 0.34 |
Clay | 0.41 |
Concrete | 0.1 – 0.2 |
Copper | 0.355 |
Cork | 0 |
Glass, Soda | 0.22 |
Glass, Float | 0.2 – 0.27 |
Granite | 0.2 – 0.3 |
Ice | 0.33 |
Inconel | 0.27 – 0.38 |
Iron, Cast – grey | 0.211 |
Iron, Cast | 0.22 – 0.30 |
Iron, Ductile | 0.26 – 0.31 |
Iron, Malleable | 0.271 |
Lead | 0.431 |
Limestone | 0.2 – 0.3 |
Magnesium | 0.35 |
Magnesium Alloy | 0.281 |
Marble | 0.2 – 0.3 |
Molybdenum | 0.307 |
Monel metal | 0.315 |
Nickel Silver | 0.322 |
Nickel Steel | 0.291 |
Polystyrene | 0.34 |
Phosphor Bronze | 0.359 |
Rubber | 0.48 – ~0.5 |
Sand | 0.29 |
Sandy loam | 0.31 |
Sandy clay | 0.37 |
Stainless Steel 18-8 | 0.305 |
Steel, cast | 0.265 |
Steel, Cold-rolled | 0.287 |
Steel, high carbon | 0.295 |
Steel, mild | 0.303 |
Titanium (99.0 Ti) | 0.32 |
Wrought iron | 0.278 |
Z-nickel | 0.36 |
Zinc | 0.331 |
Table source: https://www.engineeringtoolbox.com
Spider graph: Influences of selected glass component additions on Poisson’s ratio of a specific base glass.
FAQ
- Is Poisson’s ratio constant for a material?
As long as the material is an exact match for the one tested, then yes, it will remain constant for that material.
- Why is Poisson’s ratio important?
Understanding the effect that stress has on a material has a number of important applications, especially in mechanical engineering.
For instance, precision parts designed for aerospace engineering may need to withstand considerable stress without any serious deformation. It’s crucial that the correct material is chosen, and Poisson’s ratio will help with this choice.
Other important applications are in the design of high-pressure pipework systems, especially those that carry flammable or explosive liquids or gas.
- What is Poisson’s ratio in concrete?
The ratio for concrete varies between 0.1 for high-strength concrete, to 0.2 for weaker mixes.
- Does Poisson’s ratio have units?
No, it is considered to be dimensionless.
- Is Poisson’s ratio negative
It is positive for most materials, as a minus sign is applied to the formula to yield a positive result when the material cross-section reduces under a stretching force. Some rare materials (auxetic materials) show an increase in cross-section when stretched. These will have a negative Poisson’s ratio.
- What if Poisson’s ratio is zero?
In theory, a material that has a ratio of zero will display no transverse deformation when axial strain is applied. In practice, there are no known materials with a ratio of zero, but some get very close.
Cork shows almost zero lateral expansion upon compression, meaning the ratio is virtually zero. Other materials such as biological tissue – ligaments, cartilage, corneal and brain tissue, also approach a zero value of Poisson’s ratio.