# Poisson's ratio

Dictionary entry for Poisson's ratio (edit)

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1. REDIRECT Dictionary: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).

## Definition

${\displaystyle \mathrm {Poisson's} \ \mathrm {ratio} ={\frac {\mathrm {transverse} \ \mathrm {strain} }{\mathrm {longitudinal} \ \mathrm {strain} }}}$
${\displaystyle \nu ={\frac {\Delta W/W}{\Delta L/L}}}$

## Other expressions

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

${\displaystyle \nu ={\frac {\left({\frac {V_{\mathrm {P} }}{V_{\mathrm {S} }}}\right)^{2}-2}{2\left({\frac {V_{\mathrm {P} }}{V_{\mathrm {S} }}}\right)^{2}-2}}}$

In this case, when a material has a positive ${\displaystyle \nu }$ it will have a ${\displaystyle V_{\mathrm {P} }/V_{\mathrm {S} }}$ ratio greater than 1.42.

Expressed in terms of Lamé parameters:

${\displaystyle \nu ={\frac {\lambda }{2\,(\lambda +\mu )}}}$

## 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. Elastic constants for isotropic media expressed in terms of each other and P- and S-wave velocities (α=VP and β=VS,) and density ρ.

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 VP and VS, respectively. Following are derivations of VP in terms of VS and Poisson’s ratio s, VS in terms of VP and s, and s in terms of VS / VP where VS and VP are initially defined in terms of density r, shear modulus m and Lame’s constant l. [1]

## References

1. 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