# Difference between revisions of "Poisson's ratio"

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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). | 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). | ||

## Latest revision as of 20:10, 14 March 2015

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 , shear modulus and Lame’s constant . ^{[1]}

### Equations

By definition, Poisson's ratio:

**(**)

By definition:

**(**)

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

**(**)

**(**)

**(**)

By definition:

By definition:

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

- elastic constants for isotropic media
- elastic constants
- derivation of reflection coefficient
- derivation of Snell's Law

## External links

- Poisson's ratio — Wikipedia article