# Poisson's ratio

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

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

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

$\nu ={\frac {\left({\frac {V_{\mathrm {P} }}{V_{\mathrm {S} }}}\right)^{2}-2}{2\left({\frac {V_{\mathrm {P} }}{V_{\mathrm {S} }}}\right)^{2}-2}}$ In this case, when a material has a positive $\nu$ it will have a $V_{\mathrm {P} }/V_{\mathrm {S} }$ ratio greater than 1.42.

Expressed in terms of Lamé parameters:

$\nu ={\frac {\lambda }{2\,(\lambda +\mu )}}$ ## 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 ($\alpha =V_{p}$ and $\beta =V_{s}$ ) and density $\rho$ .

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 $V_{p}$ and $V_{s}$ , respectively. The following are derivations of $V_{p}$ in terms of $V_{s}$ and Poisson’s ratio s, $V_{s}$ in terms of $V_{p}$ and s, and s in terms of $V_{s}$ / $V_{p}$ where $V_{s}$ and $V_{p}$ are initially defined in terms of density $\rho$ , shear modulus $\mu$ and Lame’s constant $\lambda$ . 

### Equations

By definition, Poisson's ratio:

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

By definition:

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

Dividing equation 1 by equation 2,

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

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

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

By definition:

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

$V_{p}=({\frac {\lambda +2\mu }{\rho }})^{1/2}$ $V_{p}^{2}=({\frac {\lambda +2\mu }{\rho }})$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {\frac {\mu }{\rho }}{\frac {\lambda +2\mu }{\rho }}}$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {\frac {\mu }{2(\lambda +\mu )}}{\frac {\lambda +2\mu }{2(\lambda +\mu )}}}$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {\frac {\lambda +\mu -\lambda }{2(\lambda +\mu )}}{\frac {\lambda +2\mu }{2(\lambda +\mu )}}}$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {{\frac {\lambda +\mu }{2(\lambda +\mu )}}-{\frac {\lambda }{2(\lambda +\mu )}}}{\frac {\lambda +2\mu }{2(\lambda +\mu )}}}$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {{\frac {\lambda +\mu }{2(\lambda +\mu )}}-{\frac {\lambda }{2(\lambda +\mu )}}}{\frac {2(\lambda +\mu )-\lambda }{2(\lambda +\mu )}}}$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {{\frac {\lambda +\mu }{2(\lambda +\mu )}}-{\frac {\lambda }{2(\lambda +\mu )}}}{\frac {2(\lambda +\mu )-\lambda }{2(\lambda +\mu )}}}$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {0.5-{\frac {\lambda }{2(\lambda +\mu )}}}{\frac {2(\lambda +\mu )-\lambda }{2(\lambda +\mu )}}}$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {0.5-{\frac {\lambda }{2(\lambda +\mu )}}}{{\frac {2(\lambda +\mu )}{2(\lambda +\mu )}}-{\frac {\lambda }{2(\lambda +\mu )}}}}$ ${\frac {V_{s}^{2}}{V_{p}^{2}}}={\frac {0.5-{\frac {\lambda }{2(\lambda +\mu )}}}{1-{\frac {\lambda }{2(\lambda +\mu )}}}}$ By definition, Poisson's ratio:

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

Equation 6 is the same as equation 3