Difference between revisions of "Template:Elastic moduli"
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imported>Addbot m (Bot: Migrating 5 interwiki links, now provided by Wikidata on d:q11024927) |
imported>DanimothWiki (Add formulas for the calculation of E, lamda, G, nu when K and M are given.) |
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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= | + | ! colspan=12 | Conversion formulas |
|- | |- | ||
| − | | colspan= | + | | colspan=12 | 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>(K,\,\lambda)</math> | | align=center | <math>(K,\,\lambda)</math> | ||
| align=center | <math>(K,\,G)</math> | | align=center | <math>(K,\,G)</math> | ||
| − | | align=center | <math>(K,\, \nu)</math> | + | | align=center | <math>(K,\,\nu)</math> |
| + | | align=center | <math>(K,\,M)</math> | ||
| align=center | <math>(E,\,G)</math> | | align=center | <math>(E,\,G)</math> | ||
| align=center | <math>(E,\,\nu)</math> | | align=center | <math>(E,\,\nu)</math> | ||
| Line 31: | Line 32: | ||
|- | |- | ||
| align=center | <math>K=\,</math> | | align=center | <math>K=\,</math> | ||
| + | | align=center | <math>K</math> | ||
| align=center | <math>K</math> | | align=center | <math>K</math> | ||
| align=center | <math>K</math> | | align=center | <math>K</math> | ||
| Line 49: | Line 51: | ||
| align=center | <math>\tfrac{9KG}{3K+G}</math> | | align=center | <math>\tfrac{9KG}{3K+G}</math> | ||
| align=center | <math>3K(1-2\nu)\,</math> | | align=center | <math>3K(1-2\nu)\,</math> | ||
| + | | align=center | <math>\tfrac{9K(M-K)}{3K+M}</math> | ||
| align=center | <math>E</math> | | align=center | <math>E</math> | ||
| align=center | <math>E</math> | | align=center | <math>E</math> | ||
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| align=center | <math>K-\tfrac{2G}{3}</math> | | align=center | <math>K-\tfrac{2G}{3}</math> | ||
| align=center | <math>\tfrac{3K\nu}{1+\nu}</math> | | align=center | <math>\tfrac{3K\nu}{1+\nu}</math> | ||
| + | | align=center | <math>\tfrac{3K-M}{2}</math> | ||
| align=center | <math>\tfrac{G(E-2G)}{3G-E}</math> | | align=center | <math>\tfrac{G(E-2G)}{3G-E}</math> | ||
| align=center | <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math> | | align=center | <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math> | ||
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| align=center | <math>G</math> | | align=center | <math>G</math> | ||
| align=center | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math> | | align=center | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math> | ||
| + | | align=center | <math>\tfrac{3(M-K)}{4}</math> | ||
| align=center | <math>G</math> | | align=center | <math>G</math> | ||
| align=center | <math>\tfrac{E}{2(1+\nu)}</math> | | align=center | <math>\tfrac{E}{2(1+\nu)}</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>\nu</math> | | align=center | <math>\nu</math> | ||
| + | | align=center | <math>\tfrac{3K-M}{3K+M}</math> | ||
| align=center | <math>\tfrac{E}{2G}-1</math> | | align=center | <math>\tfrac{E}{2G}-1</math> | ||
| align=center | <math>\nu</math> | | align=center | <math>\nu</math> | ||
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| align=center | <math>K+\tfrac{4G}{3}</math> | | align=center | <math>K+\tfrac{4G}{3}</math> | ||
| align=center | <math>\tfrac{3K(1-\nu)}{1+\nu}</math> | | align=center | <math>\tfrac{3K(1-\nu)}{1+\nu}</math> | ||
| + | | align=center | <math>M</math> | ||
| align=center | <math>\tfrac{G(4G-E)}{3G-E}</math> | | align=center | <math>\tfrac{G(4G-E)}{3G-E}</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> | ||
Revision as of 15:44, 17 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,\,E)</math> | <math>(K,\,\lambda)</math> | <math>(K,\,G)</math> | <math>(K,\,\nu)</math> | <math>(K,\,M)</math> | <math>(E,\,G)</math> | <math>(E,\,\nu)</math> | <math>(\lambda,\,G)</math> | <math>(\lambda,\,\nu)</math> | <math>(G,\,\nu)</math> | <math>(G,\,M)</math> | |
| <math>K=\,</math> | <math>K</math> | <math>K</math> | <math>K</math> | <math>K</math> | <math>K</math> | <math>\tfrac{EG}{3(3G-E)}</math> | <math>\tfrac{E}{3(1-2\nu)}</math> | <math>\lambda+ \tfrac{2G}{3}</math> | <math>\tfrac{\lambda(1+\nu)}{3\nu}</math> | <math>\tfrac{2G(1+\nu)}{3(1-2\nu)}</math> | <math>M - \tfrac{4G}{3}</math> |
| <math>E=\, </math> | <math>E</math> | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math> | <math>\tfrac{9KG}{3K+G}</math> | <math>3K(1-2\nu)\,</math> | <math>\tfrac{9K(M-K)}{3K+M}</math> | <math>E</math> | <math>E</math> | <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math> | <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math> | <math>2G(1+\nu)\,</math> | <math>\tfrac{G(3M-4G)}{M-G}</math> |
| <math>\lambda=\,</math> | <math>\tfrac{3K(3K-E)}{9K-E}</math> | <math>\lambda</math> | <math>K-\tfrac{2G}{3}</math> | <math>\tfrac{3K\nu}{1+\nu}</math> | <math>\tfrac{3K-M}{2}</math> | <math>\tfrac{G(E-2G)}{3G-E}</math> | <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math> | <math>\lambda</math> | <math>\lambda</math> | <math>\tfrac{2 G \nu}{1-2\nu}</math> | <math>M - 2G\,</math> |
| <math>G=\, </math> | <math>\tfrac{3KE}{9K-E}</math> | <math>\tfrac{3(K-\lambda)}{2}</math> | <math>G</math> | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math> | <math>\tfrac{3(M-K)}{4}</math> | <math>G</math> | <math>\tfrac{E}{2(1+\nu)}</math> | <math>G</math> | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math> | <math>G</math> | <math>G</math> |
| <math>\nu=\,</math> | <math>\tfrac{3K-E}{6K}</math> | <math>\tfrac{\lambda}{3K-\lambda}</math> | <math>\tfrac{3K-2G}{2(3K+G)}</math> | <math>\nu</math> | <math>\tfrac{3K-M}{3K+M}</math> | <math>\tfrac{E}{2G}-1</math> | <math>\nu</math> | <math>\tfrac{\lambda}{2(\lambda + G)}</math> | <math>\nu</math> | <math>\nu</math> | <math>\tfrac{M - 2G}{2M - 2G}</math> |
| <math>M=\,</math> | <math>\tfrac{3K(3K+E)}{9K-E}</math> | <math>3K-2\lambda\,</math> | <math>K+\tfrac{4G}{3}</math> | <math>\tfrac{3K(1-\nu)}{1+\nu}</math> | <math>M</math> | <math>\tfrac{G(4G-E)}{3G-E}</math> | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math> | <math>\lambda+2G\,</math> | <math>\tfrac{\lambda(1-\nu)}{\nu}</math> | <math>\tfrac{2G(1-\nu)}{1-2\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