Difference between revisions of "Template:Elastic moduli"

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(Transpose table for easier reading and in preparation of adding more info)
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(Add Notes column)
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{| 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=7 | Conversion formulas
+
! colspan=8 | Conversion formulas
 
|-
 
|-
| colspan=7 | 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.
 
|-
 
|-
 
|
 
|
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| align=center | <math>\nu=\,</math>
 
| align=center | <math>\nu=\,</math>
 
| align=center | <math>M=\,</math>
 
| align=center | <math>M=\,</math>
 +
| align=center | Notes
  
 
|-
 
|-
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| align=center | <math>\tfrac{3K-E}{6K}</math>
 
| align=center | <math>\tfrac{3K-E}{6K}</math>
 
| align=center | <math>\tfrac{3K(3K+E)}{9K-E}</math>
 
| align=center | <math>\tfrac{3K(3K+E)}{9K-E}</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\tfrac{\lambda}{3K-\lambda}</math>
 
| align=center | <math>\tfrac{\lambda}{3K-\lambda}</math>
 
| align=center | <math>3K-2\lambda\,</math>
 
| align=center | <math>3K-2\lambda\,</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\tfrac{3K-2G}{2(3K+G)}</math>
 
| align=center | <math>\tfrac{3K-2G}{2(3K+G)}</math>
 
| align=center | <math>K+\tfrac{4G}{3}</math>
 
| align=center | <math>K+\tfrac{4G}{3}</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\nu</math>
 
| align=center | <math>\nu</math>
 
| align=center | <math>\tfrac{3K(1-\nu)}{1+\nu}</math>
 
| align=center | <math>\tfrac{3K(1-\nu)}{1+\nu}</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\tfrac{3K-M}{3K+M}</math>
 
| align=center | <math>\tfrac{3K-M}{3K+M}</math>
 
| align=center | <math>M</math>
 
| align=center | <math>M</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\tfrac{E}{2G}-1</math>
 
| align=center | <math>\tfrac{E}{2G}-1</math>
 
| align=center | <math>\tfrac{G(4G-E)}{3G-E}</math>
 
| align=center | <math>\tfrac{G(4G-E)}{3G-E}</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\nu</math>
 
| align=center | <math>\nu</math>
 
| align=center | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math>
 
| align=center | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\tfrac{\lambda}{2(\lambda + G)}</math>
 
| align=center | <math>\tfrac{\lambda}{2(\lambda + G)}</math>
 
| align=center | <math>\lambda+2G\,</math>
 
| align=center | <math>\lambda+2G\,</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\nu</math>
 
| align=center | <math>\nu</math>
 
| align=center | <math>\tfrac{\lambda(1-\nu)}{\nu}</math>
 
| align=center | <math>\tfrac{\lambda(1-\nu)}{\nu}</math>
 +
| align=center | Cannot be used when <math>\nu = 0</math>
  
 
|-
 
|-
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| align=center | <math>\tfrac{\lambda}{M+\lambda}</math>
 
| align=center | <math>\tfrac{\lambda}{M+\lambda}</math>
 
| align=center | <math>M</math>
 
| align=center | <math>M</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\nu</math>
 
| align=center | <math>\nu</math>
 
| align=center | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math>
 
| align=center | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\tfrac{M - 2G}{2M - 2G}</math>
 
| align=center | <math>\tfrac{M - 2G}{2M - 2G}</math>
 
| align=center | <math>M</math>
 
| align=center | <math>M</math>
 +
|
  
 
|-
 
|-
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| align=center | <math>\nu</math>
 
| align=center | <math>\nu</math>
 
| align=center | <math>M</math>
 
| align=center | <math>M</math>
 +
|
  
 
|}<noinclude>
 
|}<noinclude>

Revision as of 19:32, 25 May 2014

Conversion formulas
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>K</math> <math>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>K</math> <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math> <math>\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>K</math> <math>\tfrac{9KG}{3K+G}</math> <math>K-\tfrac{2G}{3}</math> <math>G</math> <math>\tfrac{3K-2G}{2(3K+G)}</math> <math>K+\tfrac{4G}{3}</math>
<math>(K,\,\nu)</math> <math>K</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>\nu</math> <math>\tfrac{3K(1-\nu)}{1+\nu}</math>
<math>(K,\,M)</math> <math>K</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>M</math>
<math>(E,\,G)</math> <math>\tfrac{EG}{3(3G-E)}</math> <math>E</math> <math>\tfrac{G(E-2G)}{3G-E}</math> <math>G</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>E</math> <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math> <math>\tfrac{E}{2(1+\nu)}</math> <math>\nu</math> <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math>
<math>(\lambda,\,G)</math> <math>\lambda+ \tfrac{2G}{3}</math> <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math> <math>\lambda</math> <math>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>\lambda</math> <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math> <math>\nu</math> <math>\tfrac{\lambda(1-\nu)}{\nu}</math> Cannot be used when <math>\nu = 0</math>
<math>(\lambda,\,M)</math> <math>\tfrac{M + 2\lambda}{3}</math> <math>\tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}</math> <math>\lambda</math> <math>\tfrac{M-\lambda}{2}</math> <math>\tfrac{\lambda}{M+\lambda}</math> <math>M</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>G</math> <math>\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>G</math> <math>\tfrac{M - 2G}{2M - 2G}</math> <math>M</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> <math>\nu</math> <math>M</math>

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

fr:Modèle:Modules élastiques