Difference between revisions of "Poisson's ratio"

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{{Definition}}
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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).
 
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).
  
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==Derivation of Poisson's ratio==
 
==Derivation of Poisson's ratio==
[[File:Sege5.jpg|thumb|Figure E-5. Elastic constants for isotropic media expressed in terms of each other and P- and S-wave velocities (&#x03B1;=''V''<sub>''P''</sub> and &#x03B2;=''V''<sub>''S''</sub>,) and density &#x03C1;.]] '''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 and Lame’s constant l. <ref name=Sheriff2002>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 http://dx.doi.org/10.1190/1.9781560802969]</ref>
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[[File:Sege5.jpg|thumb|Figure E-5. Elastic constants for isotropic media expressed in terms of each other and P- and S-wave velocities (<math>\alpha = V_{p}</math> and <math>\beta = V_{s}</math>) and density <math>\rho</math>.]] '''Figure E-5''' in Sheriff’s Encyclopedic Dictionary of Applied Geophysics contains basic information on [[Dictionary:Elastic_constants,_elastic_moduli|elastic constants]] in isotropic media expressed in terms of each other and compressional and shear wave velocities <math>V_{p}</math> and <math>V_{s}</math>, respectively.  The following are derivations of <math>V_{p}</math> in terms of <math>V_{s}</math> and Poisson’s ratio s, <math>V_{s}</math> in terms of <math>V_{p}</math> and s, and s in terms of <math>V_{s}</math> / <math>V_{p}</math> where <math>V_{s}</math> and <math>V_{p}</math> are initially defined in terms of density <math>\rho</math>, shear modulus <math>\mu</math> and Lame’s constant <math>\lambda</math>. <ref name=Sheriff2002>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 http://dx.doi.org/10.1190/1.9781560802969]</ref>
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===Equations===
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By definition, [[Poisson's ratio]]:
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{{NumBlk|:|<math>\sigma = \frac{V_{p}^{2}-2V_{s}^{2}}{2(V_{p}^{2}-V_{s}^2)}</math>
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<math>2\sigma = \frac{V_{p}^{2}-2V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}</math>
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<math>2\sigma = \frac{V_{p}^{2}-V_{s}^{2}-V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}</math>
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<math>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}}]}</math>
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<math>2\sigma = 1 - \frac{V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}</math>
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<math>2 \sigma + \frac{V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}} = 1</math>
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<math>\frac{V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}} = 1 - 2 \sigma</math>
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<math>\frac{[\frac{V_{s}^{2}}{V_{p}^{2}-V_{s}^{2}}]}{{2}} = 0.5 - \sigma</math>
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<math>\frac{{V_{s}^{2}}}{2(V_{p}^{2}-V_{s}^{2})} = 0.5 - \sigma</math>
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|{{EquationRef|1}}}}
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By definition:
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{{NumBlk|:|<math>\sigma = \frac{V_{p}^{2}-2V_{s}^{2}}{2(V_{p}^{2}-2V_{s}^{2})}</math>
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<math>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})}</math>
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<math>1 - \sigma = \frac{2V_{p}^{2}-2V_{s}^{2}-V_{p}^{2}+2V_{s}^{2}}{2(V_{p}^{2}-V_{s}^{2})}</math>
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<math>1 - \sigma = \frac{V_{p}^{2}}{2(V_{p}^{2}-V_{s}^{2})}</math>
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|{{EquationRef|2}}}}
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Dividing equation {{EquationNote|1}} by equation {{EquationNote|2}},
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{{NumBlk|:|<math>\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})}]}</math>
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<math>\frac{0.5-\sigma}{1-\sigma}=\frac{V_{s}^{2}}{V_{p}^{2}}</math>
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<math>[\frac{0.5-\sigma}{1-\sigma}]^{1/2} = \frac{V_{s}}{V_{p}}</math>
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<math>[\frac{0.5-\sigma}{1-\sigma}]^{1/2}*V_{p} = V_{s}</math>
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|{{EquationRef|3}}}}
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{{NumBlk|:|<math>V_{s}=V_{p}[\frac{0.5-\sigma}{1-\sigma}]^{1/2}</math>|{{EquationRef|4}}}}
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{{NumBlk|:|<math>V_{p}=V_{s}[\frac{0.5-\sigma}{1-\sigma}]^{1/2}</math>|{{EquationRef|5}}}}
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By definition:
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:<math>V_{s}=(\frac{\mu}{\rho})^{1/2}</math>
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:<math>V_{s}^{2}=\frac{\mu}{\rho}</math>
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By definition:
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:<math>V_{p}=(\frac{\lambda+2\mu}{\rho})^{1/2}</math>
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:<math>V_{p}^{2}=(\frac{\lambda+2\mu}{\rho})</math>
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:<math>\frac{V_{s}^{2}}{V_{p}^{2}}=\frac{\frac{\mu}{\rho}}{\frac{\lambda+2\mu}{\rho}}</math>
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:<math>\frac{V_{s}^{2}}{V_{p}^{2}}=\frac{\frac{\mu}{2(\lambda+\mu)}}{\frac{\lambda+2\mu}{2(\lambda+\mu)}}</math>
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:<math>\frac{V_{s}^{2}}{V_{p}^{2}}=\frac{\frac{\lambda+\mu-\lambda}{2(\lambda+\mu)}}{\frac{\lambda+2\mu}{2(\lambda+\mu)}}</math>
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:<math>\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)}}</math>
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:<math>\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)}}</math>
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:<math>\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)}}</math>
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:<math>\frac{V_{s}^{2}}{V_{p}^{2}}=\frac{0.5-\frac{\lambda}{2(\lambda+\mu)}}{\frac{2(\lambda+\mu)-\lambda}{2(\lambda+\mu)}}</math>
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:<math>\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)}}</math>
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:<math>\frac{V_{s}^{2}}{V_{p}^{2}}=\frac{0.5-\frac{\lambda}{2(\lambda+\mu)}}{1 - \frac{\lambda}{2(\lambda+\mu)}}</math>
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By definition, Poisson's ratio:
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{{NumBlk|:|<math>\sigma=\frac{\lambda}{2(\lambda+\mu)}</math>
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[[File:Elastic constants page1.PNG]]
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<math>\frac{V_{s}^{2}}{V_{p}^{2}}=\frac{0.5-\sigma}{1-\sigma}</math>
[[File:Elastic constants page2.PNG]]
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[[File:Elastic constants page3.PNG]]
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[[File:Elastic constants page4.PNG]]
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<math>\frac{V_{s}}{V_{p}}=[\frac{0.5-\sigma}{1-\sigma}]^{1/2}</math>
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|6}}
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Equation {{EquationNote|6}} is the same as equation {{EquationNote|3}}
  
 
==References==
 
==References==
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*[[Dictionary:Fig_E-5|elastic constants for isotropic media]]
 
*[[Dictionary:Fig_E-5|elastic constants for isotropic media]]
 
*[[Dictionary:Elastic_constants,_elastic_moduli|elastic constants]]
 
*[[Dictionary:Elastic_constants,_elastic_moduli|elastic constants]]
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*[[Reflection_coefficient|derivation of reflection coefficient]]
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*[[Snell%27s_law#Derivation_of_Snell.27s_Law|derivation of Snell's Law]]
  
 
==External links==
 
==External links==
 
* [[Wikipedia:Poisson's ratio|Poisson's ratio]] — Wikipedia article
 
* [[Wikipedia:Poisson's ratio|Poisson's ratio]] — Wikipedia article
 
{{search}}
 
{{search}}
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[[Category:derivations]]

Latest revision as of 20:10, 14 March 2015

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 , shear modulus and Lame’s constant . [1]

Equations

By definition, Poisson's ratio:











(1)


By definition:






(2)


Dividing equation 1 by equation 2,






(3)


(4)


(5)


By definition:



By definition:












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

find literature about
Poisson's ratio
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