Difference between revisions of "Template:Elastic moduli"

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{{Navbox
 
{{Navbox
| name = Elastic moduli
+
| name = Elastic moduli
 
| title = [[Elastic modulus|Elastic moduli]] for homogeneous [[isotropic]] materials
 
| title = [[Elastic modulus|Elastic moduli]] for homogeneous [[isotropic]] materials
| list1 = [[Bulk modulus]] (<math>K</math>) [[Young's modulus]] (<math>E</math>) [[Lamé parameters|Lamé's first parameter]] (<math>\lambda</math>) [[Shear modulus]] (<math>G</math>) [[Poisson's ratio]] (<math>\nu</math>) [[P-wave modulus]] (<math>M</math>)
+
|listclass = hlist
 +
| list1 =
 +
* [[Bulk modulus]] (<math>K</math>)
 +
* [[Young's modulus]] (<math>E</math>)
 +
* [[Lamé parameters|Lamé's first parameter]] (<math>\lambda</math>)
 +
* [[Shear modulus]] (<math>G, \mu</math>)
 +
* [[Poisson's ratio]] (<math>\nu</math>)
 +
* [[P-wave modulus]] (<math>M</math>)
 
| state = show
 
| state = show
 
}}
 
}}
 
{| class="wikitable collapsible" width="100%" style="font-size:smaller; background:white" align=center
 
{| class="wikitable collapsible" width="100%" style="font-size:smaller; background:white" align=center
! colspan=11 | Conversion formulas
+
! colspan=8 | Conversion formulae
 
|-
 
|-
| colspan=11 | Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.
+
| colspan=8 | Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas.
 
|-
 
|-
|  
+
|
| align=center | <math>(\lambda,\,G)</math>
+
| style="text-align:center;" | <math>K=\,</math>
| align=center | <math>(E,\,G)</math>
+
| style="text-align:center;" | <math>E=\, </math>
| align=center | <math>(K,\,\lambda)</math>
+
| style="text-align:center;" | <math>\lambda=\,</math>
| align=center | <math>(K,\,G)</math>
+
| style="text-align:center;" | <math>G=\, </math>
| align=center | <math>(\lambda,\,\nu)</math>
+
| style="text-align:center;" | <math>\nu=\,</math>
| align=center | <math>(G,\,\nu)</math>
+
| style="text-align:center;" | <math>M=\,</math>
| align=center | <math>(E,\,\nu)</math>
+
| style="text-align:center;" | Notes
| align=center | <math>(K,\, \nu)</math>
+
 
| align=center | <math>(K,\,E)</math>
 
| align=center | <math>(M,\,G)</math>
 
 
|-
 
|-
| align=center | <math>K=\,</math>  
+
| style="text-align:center;" | <math>(K,\,E)</math>
| align=center | <small><math>\lambda+ \tfrac{2G}{3}</math> </small>
+
| style="text-align:center;" |
| align=center | <math>\tfrac{EG}{3(3G-E)}</math>  
+
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{3K(3K-E)}{9K-E}</math>
 +
| style="text-align:center;" | <math>\tfrac{3KE}{9K-E}</math>
 +
| style="text-align:center;" | <math>\tfrac{3K-E}{6K}</math>
 +
| style="text-align:center;" | <math>\tfrac{3K(3K+E)}{9K-E}</math>
 
|
 
|
 +
 +
|-
 +
 +
| style="text-align:center;" | <math>(K,\,\lambda)</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{3(K-\lambda)}{2}</math>
 +
| style="text-align:center;" | <math>\tfrac{\lambda}{3K-\lambda}</math>
 +
| style="text-align:center;" | <math>3K-2\lambda\,</math>
 
|
 
|
| align=center | <math>\tfrac{\lambda(1+\nu)}{3\nu}</math>  
+
 
| align=center | <math>\tfrac{2G(1+\nu)}{3(1-2\nu)}</math>  
+
|-
| align=center | <math>\tfrac{E}{3(1-2\nu)}</math>
+
 
 +
| style="text-align:center;" | <math>(K,\,G)</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{9KG}{3K+G}</math>
 +
| style="text-align:center;" | <math>K-\tfrac{2G}{3}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{3K-2G}{2(3K+G)}</math>
 +
| style="text-align:center;" | <math>K+\tfrac{4G}{3}</math>
 
|
 
|
 +
 +
|-
 +
 +
| style="text-align:center;" | <math>(K,\,\nu)</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>3K(1-2\nu)\,</math>
 +
| style="text-align:center;" | <math>\tfrac{3K\nu}{1+\nu}</math>
 +
| style="text-align:center;" | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{3K(1-\nu)}{1+\nu}</math>
 
|
 
|
| align=center | <math>M - \tfrac{4G}{3}</math>
+
 
 
|-
 
|-
| align=center | <math>E=\, </math>  
+
 
| align=center | <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math>  
+
| style="text-align:center;" | <math>(K,\,M)</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{9K(M-K)}{3K+M}</math>
 +
| style="text-align:center;" | <math>\tfrac{3K-M}{2}</math>
 +
| style="text-align:center;" | <math>\tfrac{3(M-K)}{4}</math>
 +
| style="text-align:center;" | <math>\tfrac{3K-M}{3K+M}</math>
 +
| style="text-align:center;" |
 
|
 
|
| align=center | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math>  
+
 
| align=center | <math>\tfrac{9KG}{3K+G}</math>  
+
|-
| align=center | <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math>  
+
 
| align=center | <math>2G(1+\nu)\,</math>  
+
| style="text-align:center;" | <math>(E,\,\lambda)</math>
 +
| style="text-align:center;" | <math>\tfrac{E + 3\lambda + R}{6}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{E-3\lambda+R}{4}</math>
 +
| style="text-align:center;" | <math>\tfrac{2\lambda}{E+\lambda+R}</math>
 +
| style="text-align:center;" | <math>\tfrac{E-\lambda+R}{2}</math>
 +
| style="text-align:center;" | <math>R=\sqrt{E^2+9\lambda^2 + 2E\lambda}</math>
 +
 
 +
|-
 +
 
 +
| style="text-align:center;" | <math>(E,\,G)</math>
 +
| style="text-align:center;" | <math>\tfrac{EG}{3(3G-E)}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{G(E-2G)}{3G-E}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{E}{2G}-1</math>
 +
| style="text-align:center;" | <math>\tfrac{G(4G-E)}{3G-E}</math>
 
|
 
|
| align=center | <math>3K(1-2\nu)\,</math>  
+
 
 +
|-
 +
 
 +
| style="text-align:center;" | <math>(E,\,\nu)</math>
 +
| style="text-align:center;" | <math>\tfrac{E}{3(1-2\nu)}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math>
 +
| style="text-align:center;" | <math>\tfrac{E}{2(1+\nu)}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math>
 
|
 
|
| align=center | <math>\tfrac{G(3M-4G)}{M-G}</math>
+
 
 
|-
 
|-
| align=center | <math>\lambda=\,</math>  
+
 
 +
| style="text-align:center;" | <math>(E,\,M)</math>
 +
| style="text-align:center;" | <math>\tfrac{3M-E+S}{6}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{M-E+S}{4}</math>
 +
| style="text-align:center;" | <math>\tfrac{3M+E-S}{8}</math>
 +
| style="text-align:center;" | <math>\tfrac{E-M+S}{4M}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>S=\pm\sqrt{E^2+9M^2-10EM}</math><br />
 +
There are two valid solutions.<br />
 +
The plus sign leads to <math>\nu\geq 0</math>.<br />
 +
The minus sign leads to <math>\nu\leq 0</math>.<br /></p>
 +
 
 +
|-
 +
 
 +
| style="text-align:center;" | <math>(\lambda,\,G)</math>
 +
| style="text-align:center;" | <math>\lambda+ \tfrac{2G}{3}</math>
 +
| style="text-align:center;" | <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{\lambda}{2(\lambda + G)}</math>
 +
| style="text-align:center;" | <math>\lambda+2G\,</math>
 
|
 
|
| align=center | <math>\tfrac{G(E-2G)}{3G-E}</math>
+
 
|
+
|-
| align=center | <math>K-\tfrac{2G}{3}</math>
+
 
|  
+
| style="text-align:center;" | <math>(\lambda,\,\nu)</math>
| align=center | <math>\tfrac{2 G \nu}{1-2\nu}</math>  
+
| style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)}{3\nu}</math>
| align=center | <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math>
+
| style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math>
| align=center | <math>\tfrac{3K\nu}{1+\nu}</math>  
+
| style="text-align:center;" |
| align=center | <math>\tfrac{3K(3K-E)}{9K-E}</math>  
+
| style="text-align:center;" | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math>
| align=center | <math>M - 2G\,</math>
+
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{\lambda(1-\nu)}{\nu}</math>
 +
| style="text-align:center;" | Cannot be used when <math>\nu=0 \Leftrightarrow \lambda=0</math>
 +
 
 
|-
 
|-
| align=center | <math>G=\, </math>  
+
 
 +
| style="text-align:center;" | <math>(\lambda,\,M)</math>
 +
| style="text-align:center;" | <math>\tfrac{M + 2\lambda}{3}</math>
 +
| style="text-align:center;" | <math>\tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{M-\lambda}{2}</math>
 +
| style="text-align:center;" | <math>\tfrac{\lambda}{M+\lambda}</math>
 +
| style="text-align:center;" |
 
|
 
|
|
+
 
| align=center | <math>\tfrac{3(K-\lambda)}{2}</math>
 
|
 
| align=center | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math>
 
|
 
| align=center | <math>\tfrac{E}{2(1+\nu)}</math>
 
| align=center | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math>
 
| align=center | <math>\tfrac{3KE}{9K-E}</math>
 
 
|-
 
|-
| align=center | <math>\nu=\,</math>  
+
 
| align=center | <math>\tfrac{\lambda}{2(\lambda + G)}</math>  
+
| style="text-align:center;" | <math>(G,\,\nu)</math>
| align=center | <math>\tfrac{E}{2G}-1</math>  
+
| style="text-align:center;" | <math>\tfrac{2G(1+\nu)}{3(1-2\nu)}</math>
| align=center | <math>\tfrac{\lambda}{3K-\lambda}</math>
+
| style="text-align:center;" | <math>2G(1+\nu)\,</math>
| align=center | <math>\tfrac{3K-2G}{2(3K+G)}</math>  
+
| style="text-align:center;" | <math>\tfrac{2 G \nu}{1-2\nu}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math>
 
|
 
|
|
+
 
|
 
|
 
| align=center | <math>\tfrac{3K-E}{6K}</math>
 
| align=center | <math>\tfrac{M - 2G}{2M - 2G}</math>
 
 
|-
 
|-
| align=center | <math>M=\,</math>
 
| align=center | <math>\lambda+2G\,</math>
 
| align=center | <math>\tfrac{G(4G-E)}{3G-E}</math>
 
| align=center | <math>3K-2\lambda\,</math>
 
| align=center | <math>K+\tfrac{4G}{3}</math>
 
| align=center | <math>\tfrac{\lambda(1-\nu)}{\nu}</math>
 
| align=center | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math>
 
| align=center | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math>
 
| align=center | <math>\tfrac{3K(1-\nu)}{1+\nu}</math>
 
| align=center | <math>\tfrac{3K(3K+E)}{9K-E}</math>
 
|}<noinclude>
 
  
The stiffness matrix (9 by 9, or 6 by 6 in [[Voigt notation]]) in [[Hooke's law]] (in 3D) can be parametrized by only two components for homogeneous and isotropic materials. One may choose whichever pair one prefers among the elastic moduli given below. Some of the possible conversions are listed in the table.
+
| style="text-align:center;" | <math>(G,\,M)</math>
 +
| style="text-align:center;" | <math>M - \tfrac{4G}{3}</math>
 +
| style="text-align:center;" | <math>\tfrac{G(3M-4G)}{M-G}</math>
 +
| style="text-align:center;" | <math>M - 2G\,</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" | <math>\tfrac{M - 2G}{2M - 2G}</math>
 +
| style="text-align:center;" |
 +
|
  
 +
|-
  
==References==
+
| style="text-align:center;" | <math>(\nu,\,M)</math>
* G. Mavko, T. Mukerji, J. Dvorkin. ''The Rock Physics Handbook''. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4
+
| style="text-align:center;" | <math>\tfrac{M(1+\nu)}{3(1-\nu)}</math>
 +
| style="text-align:center;" | <math>\tfrac{M(1+\nu)(1-2\nu)}{1-\nu}</math>
 +
| style="text-align:center;" | <math>\tfrac{M \nu}{1-\nu}</math>
 +
| style="text-align:center;" | <math>\tfrac{M(1-2\nu)}{2(1-\nu)}</math>
 +
| style="text-align:center;" |
 +
| style="text-align:center;" |
 +
|
  
[[Category:Continuum mechanics|{{PAGENAME}}]]
+
|}<noinclude>
[[Category:Physics templates|{{PAGENAME}}]]
+
{{Documentation}}
 
+
</noinclude>
[[da:Skabelon:Parametre for materialers elasticitet]]
 
[[eo:Ŝablono:Elastaj moduloj]]
 
[[es:Plantilla:Módulo de elasticidad]]
 
[[fa:الگو:مدول‌های کشسانی]]
 
</include>
 

Latest revision as of 23:16, 1 February 2019

Conversion formulae
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas.
<math>K=\,</math> <math>E=\, </math> <math>\lambda=\,</math> <math>G=\, </math> <math>\nu=\,</math> <math>M=\,</math> Notes
<math>(K,\,E)</math> <math>\tfrac{3K(3K-E)}{9K-E}</math> <math>\tfrac{3KE}{9K-E}</math> <math>\tfrac{3K-E}{6K}</math> <math>\tfrac{3K(3K+E)}{9K-E}</math>
<math>(K,\,\lambda)</math> <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math> <math>\tfrac{3(K-\lambda)}{2}</math> <math>\tfrac{\lambda}{3K-\lambda}</math> <math>3K-2\lambda\,</math>
<math>(K,\,G)</math> <math>\tfrac{9KG}{3K+G}</math> <math>K-\tfrac{2G}{3}</math> <math>\tfrac{3K-2G}{2(3K+G)}</math> <math>K+\tfrac{4G}{3}</math>
<math>(K,\,\nu)</math> <math>3K(1-2\nu)\,</math> <math>\tfrac{3K\nu}{1+\nu}</math> <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math> <math>\tfrac{3K(1-\nu)}{1+\nu}</math>
<math>(K,\,M)</math> <math>\tfrac{9K(M-K)}{3K+M}</math> <math>\tfrac{3K-M}{2}</math> <math>\tfrac{3(M-K)}{4}</math> <math>\tfrac{3K-M}{3K+M}</math>
<math>(E,\,\lambda)</math> <math>\tfrac{E + 3\lambda + R}{6}</math> <math>\tfrac{E-3\lambda+R}{4}</math> <math>\tfrac{2\lambda}{E+\lambda+R}</math> <math>\tfrac{E-\lambda+R}{2}</math> <math>R=\sqrt{E^2+9\lambda^2 + 2E\lambda}</math>
<math>(E,\,G)</math> <math>\tfrac{EG}{3(3G-E)}</math> <math>\tfrac{G(E-2G)}{3G-E}</math> <math>\tfrac{E}{2G}-1</math> <math>\tfrac{G(4G-E)}{3G-E}</math>
<math>(E,\,\nu)</math> <math>\tfrac{E}{3(1-2\nu)}</math> <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math> <math>\tfrac{E}{2(1+\nu)}</math> <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math>
<math>(E,\,M)</math> <math>\tfrac{3M-E+S}{6}</math> <math>\tfrac{M-E+S}{4}</math> <math>\tfrac{3M+E-S}{8}</math> <math>\tfrac{E-M+S}{4M}</math> <math>S=\pm\sqrt{E^2+9M^2-10EM}</math>

There are two valid solutions.
The plus sign leads to <math>\nu\geq 0</math>.

The minus sign leads to <math>\nu\leq 0</math>.

<math>(\lambda,\,G)</math> <math>\lambda+ \tfrac{2G}{3}</math> <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math> <math>\tfrac{\lambda}{2(\lambda + G)}</math> <math>\lambda+2G\,</math>
<math>(\lambda,\,\nu)</math> <math>\tfrac{\lambda(1+\nu)}{3\nu}</math> <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math> <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math> <math>\tfrac{\lambda(1-\nu)}{\nu}</math> Cannot be used when <math>\nu=0 \Leftrightarrow \lambda=0</math>
<math>(\lambda,\,M)</math> <math>\tfrac{M + 2\lambda}{3}</math> <math>\tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}</math> <math>\tfrac{M-\lambda}{2}</math> <math>\tfrac{\lambda}{M+\lambda}</math>
<math>(G,\,\nu)</math> <math>\tfrac{2G(1+\nu)}{3(1-2\nu)}</math> <math>2G(1+\nu)\,</math> <math>\tfrac{2 G \nu}{1-2\nu}</math> <math>\tfrac{2G(1-\nu)}{1-2\nu} </math>
<math>(G,\,M)</math> <math>M - \tfrac{4G}{3}</math> <math>\tfrac{G(3M-4G)}{M-G}</math> <math>M - 2G\,</math> <math>\tfrac{M - 2G}{2M - 2G}</math>
<math>(\nu,\,M)</math> <math>\tfrac{M(1+\nu)}{3(1-\nu)}</math> <math>\tfrac{M(1+\nu)(1-2\nu)}{1-\nu}</math> <math>\tfrac{M \nu}{1-\nu}</math> <math>\tfrac{M(1-2\nu)}{2(1-\nu)}</math>
Template documentation[view] [edit] [history] [purge]

Usage

The stiffness matrix (9 by 9, or 6 by 6 in Voigt notation) in Hooke's law (in 3D) can be parametrized by only two components for homogeneous and isotropic materials. One may choose whichever pair one prefers among the elastic moduli given below. Some of the possible conversions are listed in the table.

References

  • G. Mavko, T. Mukerji, J. Dvorkin. The Rock Physics Handbook. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4