# Difference between revisions of "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 ${\displaystyle V_{p}}$ and ${\displaystyle V_{s}}$, respectively. Following are derivations of ${\displaystyle V_{p}}$ in terms of ${\displaystyle V_{s}}$ and Poisson’s ratio s, ${\displaystyle V_{s}}$ in terms of ${\displaystyle V_{p}}$ and s, and s in terms of ${\displaystyle V_{s}}$ / ${\displaystyle V_{p}}$ where ${\displaystyle V_{s}}$ and ${\displaystyle V_{p}}$ are initially defined in terms of density ${\displaystyle \rho }$, shear modulus ${\displaystyle \mu }$ and Lame’s constant ${\displaystyle \lambda }$. [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