Poisson's Ratio Calculator
Find Poisson's ratio from the elastic moduli of a material, and the sideways strain that goes with a pull.
What it does
Pull a rubber band and it gets thinner as well as longer. That is not a quirk of rubber. Every solid you stretch does it, and the size of the effect is a constant of the material called Poisson's ratio.
OpenStax University Physics sets it out in the elasticity section. A material under stress deforms, and the relation between the stress and the strain is an elastic modulus, which is a measured laboratory number: Young's modulus for tension, the shear modulus for sideways stress and the bulk modulus for squeezing from all sides. Each of those is different for each material, and the values in the tables are quite far apart, so they are worth checking against each other.
Poisson's ratio ties them together. Because a material has to conserve volume while it is being stretched, the sideways strain always has the opposite sign to the lengthwise one, and the ratio between them is fixed for a given material. The elastic moduli are not independent either: for an isotropic solid they follow from two numbers, and Poisson's ratio is the link between them.
This page works that link in both directions. Give it Young's modulus and the shear modulus and it returns Poisson's ratio, and give it Young's modulus and the bulk modulus and it returns the same ratio the other way round, so you can check a table of handbook values against itself. It also gives the transverse strain that goes with an axial strain, and the energy stored per unit volume when the material is stretched.
The comparison of the two routes is the useful part. Real materials are not perfectly isotropic, and they behave differently in tension and in compression, so the numbers in a handbook rarely satisfy both relations exactly. Aluminium in the standard table has a Young's modulus of 70 GPa, a shear modulus of 25 GPa and a bulk modulus of 75 GPa, and the two ways of getting Poisson's ratio from them give 0.4000 and 0.3444. A gap that size is normal for rounded data, and a much larger gap means you have mixed numbers from different sources.
How to use
- Choose the units for the moduli, gigapascals or thousands of pounds per square inch.
- Enter the Young's modulus of the material, the one for tension and compression.
- Enter the shear modulus, for sideways stress.
- Enter the bulk modulus, for squeezing from all sides.
- Read Poisson's ratio from the first two moduli, and compare it with the value from the Young's and bulk pair.
- Enter the axial strain you are interested in and read the transverse strain and the stored energy.
Formula
- nu
- Poisson's ratio of the material, a number usually between 0 and 0.5.
- E
- Young's modulus, the elastic modulus for tension and compression.
- G
- Shear modulus, the elastic modulus for sideways stress.
- B
- Bulk modulus, the elastic modulus for pressure from all sides.
- strain
- Change in length divided by the original length, with no unit.
The two formulas for Poisson's ratio are equivalent for a perfectly isotropic material, so any difference between them comes from the data, not the physics. A Poisson's ratio of 0.5 is the limit for an incompressible material, where a squeeze in one direction is exactly balanced by a bulge in the others, and rubber sits near 0.49. A negative ratio is possible in a material with auxetic behaviour, where pulling makes the sample fatter, though no common material does that.
Worked example
Example: aluminium with a strain of 0.001
Defaults: metric units, a Young's modulus of 70 GPa, a shear modulus of 25 GPa, a bulk modulus of 75 GPa and an axial strain of 0.001.
- From Young's and shear modulus: nu = 70 / (2 x 25) - 1 = 1.4 - 1 = 0.4000.
- From Young's and bulk modulus: nu = (1 - 70 / (3 x 75)) / 2 = (1 - 0.31111) / 2 = 0.3444.
- The two values differ by 0.0556 because the handbook numbers are rounded, and the true ratio for aluminium is close to 0.35.
- Transverse strain: -0.4000 x 0.001 = -0.000400, so the sample narrows as it is pulled.
- Energy stored per unit volume: 1/2 x 70e9 x 0.001 squared = 35,000 J per cubic metre.
- The tool shows 0.4000, 0.3444, -0.000400 and 35,000.00 J per m3.
Reference table
| Material | Young's modulus | Shear modulus | Bulk modulus | Poisson's ratio from E and G |
|---|---|---|---|---|
| Aluminium | 70 GPa | 25 GPa | 75 GPa | 0.4000 |
| Brass | 90 GPa | 35 GPa | 60 GPa | 0.2857 |
| Copper | 110 GPa | 44 GPa | 140 GPa | 0.2500 |
| Crown glass | 60 GPa | 25 GPa | 50 GPa | 0.2000 |
| Granite | 45 GPa | 20 GPa | 45 GPa | 0.1250 |
Limits
- The relations assume an isotropic material, so a material that behaves differently in different directions, such as wood or a composite, will not satisfy them.
- The values are small strain values, and metals start to behave differently once the stress is high enough to leave the linear range.
- Only elastic behaviour is covered, so nothing here describes creep, yielding or what happens past the elastic limit.
- The two routes to Poisson's ratio use handbook values, so the difference between them measures the rounding of the data rather than anything about the material.
Common mistakes
- Mixing moduli from different tables. The result is a Poisson's ratio that belongs to no material at all.
- Forgetting that the bulk modulus appears with a factor of three, since pressure acts on every surface at once.
- Expecting the transverse strain to have the same sign as the axial strain. The two always have opposite signs.
- Using the relation for an incompressible material on a solid. A Poisson's ratio of 0.5 belongs to liquids, not to steel.
- Reading a large gap between the two Poisson's ratio values as a physical effect. It is a data quality warning.
Try it yourself
Quick checks. Answers sit tucked under each question.
What is Poisson's ratio for a material with a Young's modulus of 70 GPa and a shear modulus of 25 GPa?
0.4000, from 70 / (2 x 25) - 1 = 1.4 - 1.
A rod is pulled to a strain of 0.002 and the material has a Poisson's ratio of 0.3. What is the sideways strain?
-0.0006, because the transverse strain is -0.3 x 0.002. The rod narrows as it is stretched.
How much energy per unit volume is stored in a material with a Young's modulus of 70 GPa at a strain of 0.001?
35,000 J per cubic metre, from one half times 70e9 times 0.001 squared.
Questions
What is Poisson's ratio?
The negative ratio of the sideways strain to the lengthwise strain in a stretched material. It is a constant of the material, usually between 0 and 0.5.
Why does a stretched material get thinner?
Because the material has to keep its volume, roughly. Pull it in one direction and it has to give way in the others, which is why Poisson's ratio is never negative for an ordinary solid.
What is the limit for Poisson's ratio?
0.5, which corresponds to a perfectly incompressible material. Rubber sits just below it at about 0.49 and liquids are at 0.5.
Which moduli are independent?
Only two of the three. For an isotropic solid, Poisson's ratio and one elastic modulus fix the other two, which is why the page can check a table against itself.
Why do the two answers for aluminium differ?
Because the handbook values are rounded to one or two significant figures. The true Poisson's ratio of aluminium is about 0.35, and both routes are approximations to it.
Does this work for wood?
Not well. Wood is anisotropic and its moduli differ along the grain and across it, so a single Poisson's ratio does not describe it.
Sources
- OpenStax University Physics Volume 1, 12.3: stress, strain and elastic modulus
- OpenStax College Physics 2e, 5.3: elasticity, Young's modulus, shear modulus and bulk modulus
- Physics LibreTexts: table of elastic moduli for common materials