# Poisson's ratio

Dictionary entry for Poisson's ratio (edit) |
---|

<translate> - REDIRECT Dictionary:Poisson’s_ratio
</translate> |

An elastic parameter: the ratio of transverse contractional strain to longitudinal extensional strain. In other words, a measure of the degree to which a material expands outwards when squeezed, or equivalently contracts when stretched (though some materials, called *auxetic*, do display the opposite behaviour).

## Contents

## Definition

## Other expressions

Expressed in terms of acoustic velocities, assuming the material is isotropic and homogenous:

In this case, when a material has a positive it will have a ratio greater than 1.42.

Expressed in terms of Lamé parameters:

## Typical values

For incompressible material, `ν` is approximately 0.5. Cork has a value of about 0, meaning that it does not expand radially as it is compressed. Most rocks have `ν` between about 0.1 and 0.4.

Materials with negative Poisson's ratio, meaning that they get thinner as they are compressed, do exist. They are called *auxetic* and include the mineral α-cristobalite.

## Derivation of Poisson's ratio

**Figure E-5** in Sheriff’s Encyclopedic Dictionary of Applied Geophysics contains basic information on elastic constants in isotropic media expressed in terms of each other and compressional and shear wave velocities and , respectively. The following are derivations of in terms of and Poisson’s ratio s, in terms of and s, and s in terms of / where and are initially defined in terms of density **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho}**
, shear modulus **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu}**
and Lame’s constant **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda}**
. ^{[1]}

### Equations

By definition, Poisson's ratio:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2\sigma = 1 - \frac{V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}**

**(**)

By definition:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1 - \sigma = \frac{2(V_{p}^{2}-V_{s}^{2})}{2(V_{p}^{2}-V_{s}^{2})} - \frac{V_{p}^{2}-2V_{s}^{2}}{2(V_{p}^{2}-V_{s}^{2})}}**

**(**)

Dividing equation **1** by equation **2**,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{0.5-\sigma}{1-\sigma} = \frac{[\frac{V_{s}^{2}}{2(V_{p}^{2}-V_{s}^{2})}]}{[\frac{V_{p}^{2}}{2(V_{p}^{2}-V_{s}^{2})}]}}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{0.5-\sigma}{1-\sigma}=\frac{V_{s}^{2}}{V_{p}^{2}}}**

**(**)

**(**)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{p}=V_{s}[\frac{0.5-\sigma}{1-\sigma}]^{1/2}}****(**)

By definition:

By definition:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{p}=(\frac{\lambda+2\mu}{\rho})^{1/2}}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{V_{s}^{2}}{V_{p}^{2}}=\frac{\frac{\lambda+\mu}{2(\lambda+\mu)}- \frac{\lambda}{2(\lambda+\mu)}}{\frac{2(\lambda+\mu)-\lambda}{2(\lambda+\mu)}}}**

By definition, Poisson's ratio:

**(6)**

Equation **6** is the same as equation **3**

## References

- ↑ Sheriff, Robert E. (2002). Encyclopedic Dictionary of Exploration Geophysics (4th ed.). Society of Exploration Geophysicists, SEG Geophysical Reference Series No. 13. ISBN 978-1-56080-118-4. DOI: http://dx.doi.org/10.1190/1.9781560802969

## See also

## External links

- Poisson's ratio — Wikipedia article