# Poisson’s Ratio

Poisson’s ratio (symbol ν or nu) or is a mechanical property of a material that is the measure of its deformation perpendicular to the direction of loading. In other words, it is the negative value of the ratio of transverse or lateral strain (x-direction) to axial or longitudinal strain (y-direction).

The ratio takes its name from French physicist and mathematician Siméon Poisson. It measures the Poisson effect, which is the tendency of a material to compress in the direction perpendicular to the applied force. For example, a rubber band becomes thinner when you stretch it or thicker when you compress it.

In simple terms, Poisson’s ratio measures a materials ability to thicken when compressed or thin when stretched.

### Formula for Poisson’s Ratio

The reason the formula for Poisson’s ratio contains a negative sign is so the unitless scalar value is positive for most materials under tensile deformation. There are multiple ways of writing the formula:

Poisson’s ratio (ν) = transverse strain / longitudinal strain = – dεtrans/dεaxial = – dεlat/dεlong

Here, strain ε is the change in length divided by original length:

ε = ΔL/L

### What Poisson’s Ratio Means

Most materials have Poisson’s ratio values ranging between 0.0 and 0.5.

Soft materials, like rubber, have Poisson’s ratio values near 0.5. Steel and rigid polymers typically have values around 0.3. Porous materials typically have Poisson’s ratio values close to zero. This is because they collapse when compressed.

Auxetic materials have negative Poisson’s ratio values. In other words, they thicken when stretched and thin when compressed. Origami-folded materials and certain crystals are auxetic materials.

### Is Poisson’s Ratio a Constant for a Material?

Poisson’s ratio is relatively constant within elastic limits for most materials. There are exceptions.

Poisson’s ratio is not necessarily the same for elongation versus compression. Also, the ratio may be isotropic or anisotropic. For a linear isotropic material, the deformation is the same regardless of the material’s axis. For an anisotropic material, Poisson’s ratio depends on the stress and strain axis.

Usually, Poisson’s ratio gradually increases with temperature, up to a certain peak temperature. In most cases, the overall effect is small because changing temperature affects both transverse and axial strain.

### Table of Poisson’s Ratio Values

This table lists representative Poisson’s ratio values for a range of materials:

### Auxetic Materials

Auxetic materials or auxetics display a negative Poisson’s ratio. They thicken when stretched. Examples of auxetics include certain molecules, macromolecules, bulk materials, and crystals. Usually, auxetics have high fracture resistance and energy absorption. This makes them useful as shock absorbers, packing materials, and body armor.

Here are some examples of auxetic materials:

• Crystalline Li, Na, K, Cu, Fe, and many other metallic elements
• Graphene
• Living bone (probably)
• Tendons
• Most types of paper
• Some types of polytetrafluoroethylene (e.g., Gore-Tex)

### References

• Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. (1993). Advanced Mechanics of Materials. Wiley.
• Epishin, A.I.; Lisovenko, D.S. (2016). “Extreme values of Poisson’s ratio of cubic crystals”. Technical Physics. 61 (10): 1516–1524. doi:10.1016/j.mechmat.2019.03.017
• Gercek, H. (January 2007). “Poisson’s ratio values for rocks”. International Journal of Rock Mechanics and Mining Sciences. 44 (1): 1–13. doi:10.1016/j.ijrmms.2006.04.011
• Jastrzebski, D. (1959). Nature and Properties of Engineering Materials (Wiley International ed.). John Wiley & Sons, Inc.
• Lakes, R.S. (1987). “Foam structures with a negative Poisson’s ratio”. Science. 235 (4792): 1038–40. doi:10.1126/science.235.4792.1038