Poisson's ratio

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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

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. Elastic constants for isotropic media expressed in terms of each other and P- and S-wave velocities ( and ) and density .

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho} , shear modulus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu} and Lame’s constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} . [1]

Equations

By definition, Poisson's ratio:






Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2\sigma = 1 - \frac{V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}}






(1)


By definition:



Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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})}}




(2)


Dividing equation 1 by equation 2,


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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})}]}}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{0.5-\sigma}{1-\sigma}=\frac{V_{s}^{2}}{V_{p}^{2}}}




(3)


(4)


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{p}=V_{s}[\frac{0.5-\sigma}{1-\sigma}]^{1/2}} (5)


By definition:



By definition:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{p}=(\frac{\lambda+2\mu}{\rho})^{1/2}}







Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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)}}}






By definition, Poisson's ratio:




(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

See also

External links

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Poisson's ratio
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