# Poisson's ratio: Difference between revisions

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 (${\displaystyle \alpha =V_{p}}$ and ${\displaystyle \beta =V_{s}}$) and density ${\displaystyle \rho }$.

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. The 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]

### Equations

By definition, Poisson's ratio:

 ${\displaystyle \sigma ={\frac {V_{p}^{2}-2V_{s}^{2}}{2(V_{p}^{2}-V_{s}^{2})}}}$ ${\displaystyle 2\sigma ={\frac {V_{p}^{2}-2V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}}$ ${\displaystyle 2\sigma ={\frac {V_{p}^{2}-V_{s}^{2}-V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}}$ ${\displaystyle 2\sigma ={\frac {[{\frac {V_{p}^{2}-V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}]}{[{\frac {V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}]}}}$ ${\displaystyle 2\sigma =1-{\frac {V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}}$ ${\displaystyle 2\sigma +{\frac {V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}=1}$ ${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}=1-2\sigma }$ ${\displaystyle {\frac {[{\frac {V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}]}{2}}=0.5-\sigma }$ ${\displaystyle {\frac {V_{s}^{2}}{2(V_{p}^{2}-V_{s}^{2})}}=0.5-\sigma }$ (1)

By definition:

 ${\displaystyle \sigma ={\frac {V_{p}^{2}-2V_{s}^{2}}{2(V_{p}^{2}-2V_{s}^{2})}}}$ ${\displaystyle 1-\sigma ={\frac {2(V_{p}^{2}-V_{s}^{2})}{2(V_{p}^{2}-V_{s}^{2})}}-{\frac {V_{p}^{2}-2V_{s}^{2}}{2(V_{p}^{2}-V_{s}^{2})}}}$ ${\displaystyle 1-\sigma ={\frac {2V_{p}^{2}-2V_{s}^{2}-V_{p}^{2}+2V_{s}^{2}}{2(V_{p}^{2}-V_{s}^{2})}}}$ ${\displaystyle 1-\sigma ={\frac {V_{p}^{2}}{2(V_{p}^{2}-V_{s}^{2})}}}$ (2)

Dividing equation 1 by equation 2,

 ${\displaystyle {\frac {0.5-\sigma }{1-\sigma }}={\frac {[{\frac {V_{s}^{2}}{2(V_{p}^{2}-V_{s}^{2})}}]}{[{\frac {V_{p}^{2}}{2(V_{p}^{2}-V_{s}^{2})}}]}}}$ ${\displaystyle {\frac {0.5-\sigma }{1-\sigma }}={\frac {V_{s}^{2}}{V_{p}^{2}}}}$ ${\displaystyle [{\frac {0.5-\sigma }{1-\sigma }}]^{1/2}={\frac {V_{s}}{V_{p}}}}$ ${\displaystyle [{\frac {0.5-\sigma }{1-\sigma }}]^{1/2}*V_{p}=V_{s}}$ (3)

 ${\displaystyle V_{s}=V_{p}[{\frac {0.5-\sigma }{1-\sigma }}]^{1/2}}$ (4)

 ${\displaystyle V_{p}=V_{s}[{\frac {0.5-\sigma }{1-\sigma }}]^{1/2}}$ (5)

By definition:

${\displaystyle V_{s}=({\frac {\mu }{\rho }})^{1/2}}$

${\displaystyle V_{s}^{2}={\frac {\mu }{\rho }}}$

By definition:

${\displaystyle V_{p}=({\frac {\lambda +2\mu }{\rho }})^{1/2}}$

${\displaystyle V_{p}^{2}=({\frac {\lambda +2\mu }{\rho }})}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {\frac {\mu }{\rho }}{\frac {\lambda +2\mu }{\rho }}}}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {\frac {\mu }{2(\lambda +\mu )}}{\frac {\lambda +2\mu }{2(\lambda +\mu )}}}}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {\frac {\lambda +\mu -\lambda }{2(\lambda +\mu )}}{\frac {\lambda +2\mu }{2(\lambda +\mu )}}}}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {{\frac {\lambda +\mu }{2(\lambda +\mu )}}-{\frac {\lambda }{2(\lambda +\mu )}}}{\frac {\lambda +2\mu }{2(\lambda +\mu )}}}}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {{\frac {\lambda +\mu }{2(\lambda +\mu )}}-{\frac {\lambda }{2(\lambda +\mu )}}}{\frac {2(\lambda +\mu )-\lambda }{2(\lambda +\mu )}}}}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {{\frac {\lambda +\mu }{2(\lambda +\mu )}}-{\frac {\lambda }{2(\lambda +\mu )}}}{\frac {2(\lambda +\mu )-\lambda }{2(\lambda +\mu )}}}}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {0.5-{\frac {\lambda }{2(\lambda +\mu )}}}{\frac {2(\lambda +\mu )-\lambda }{2(\lambda +\mu )}}}}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {0.5-{\frac {\lambda }{2(\lambda +\mu )}}}{{\frac {2(\lambda +\mu )}{2(\lambda +\mu )}}-{\frac {\lambda }{2(\lambda +\mu )}}}}}$

${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {0.5-{\frac {\lambda }{2(\lambda +\mu )}}}{1-{\frac {\lambda }{2(\lambda +\mu )}}}}}$

By definition, Poisson's ratio:

 ${\displaystyle \sigma ={\frac {\lambda }{2(\lambda +\mu )}}}$ ${\displaystyle {\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {0.5-\sigma }{1-\sigma }}}$ ${\displaystyle {\frac {V_{s}}{V_{p}}}=[{\frac {0.5-\sigma }{1-\sigma }}]^{1/2}}$ (6)

Equation 6 is the same as equation 3

## 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