Difference between revisions of "Template:Elastic moduli"
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(Added empty boxes for G and M in last column.) |
imported>FMasic (Reordered columns systematically into list/row order.) |
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| 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=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. | ||
|- | |- | ||
| − | | | + | | |
| − | | align=center | <math>( | + | | align=center | <math>(K,\,E)</math> |
| − | |||
| 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>(E,\,G)</math> | ||
| + | | align=center | <math>(E,\,\nu)</math> | ||
| + | | align=center | <math>(\lambda,\,G)</math> | ||
| align=center | <math>(\lambda,\,\nu)</math> | | align=center | <math>(\lambda,\,\nu)</math> | ||
| align=center | <math>(G,\,\nu)</math> | | align=center | <math>(G,\,\nu)</math> | ||
| − | | align=center | <math>( | + | | align=center | <math>(G,\,M)</math> |
| − | + | ||
| − | |||
| − | |||
|- | |- | ||
| − | | align=center | <math>K=\, | + | | align=center | <math>K=\,</math> |
| − | |||
| − | |||
| | | | ||
| | | | ||
| − | |||
| − | |||
| − | |||
| | | | ||
| | | | ||
| + | | align=center | <math>\tfrac{EG}{3(3G-E)}</math> | ||
| + | | align=center | <math>\tfrac{E}{3(1-2\nu)}</math> | ||
| + | | align=center | <math>\lambda+ \tfrac{2G}{3}</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>M - \tfrac{4G}{3}</math> | | align=center | <math>M - \tfrac{4G}{3}</math> | ||
| + | |||
|- | |- | ||
| − | | align=center | <math>E=\, | + | |
| − | + | | align=center | <math>E=\, </math> | |
| | | | ||
| − | | align=center | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math> | + | | align=center | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math> |
| − | | align=center | <math>\tfrac{9KG}{3K+G}</math> | + | | align=center | <math>\tfrac{9KG}{3K+G}</math> |
| − | | align=center | <math> | + | | align=center | <math>3K(1-2\nu)\,</math> |
| − | |||
| | | | ||
| − | |||
| | | | ||
| + | | align=center | <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math> | ||
| + | | align=center | <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math> | ||
| + | | align=center | <math>2G(1+\nu)\,</math> | ||
| align=center | <math>\tfrac{G(3M-4G)}{M-G}</math> | | align=center | <math>\tfrac{G(3M-4G)}{M-G}</math> | ||
| + | |||
|- | |- | ||
| − | | align=center | <math>\lambda=\,</math> | + | |
| + | | align=center | <math>\lambda=\,</math> | ||
| + | | align=center | <math>\tfrac{3K(3K-E)}{9K-E}</math> | ||
| | | | ||
| − | |||
| − | |||
| 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{ | + | | 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> | ||
| − | | align=center | <math>\tfrac{ | + | | |
| − | + | | | |
| + | | align=center | <math>\tfrac{2 G \nu}{1-2\nu}</math> | ||
| align=center | <math>M - 2G\,</math> | | align=center | <math>M - 2G\,</math> | ||
| + | |||
|- | |- | ||
| − | | align=center | <math>G=\, </math> | + | |
| + | | align=center | <math>G=\, </math> | ||
| + | | align=center | <math>\tfrac{3KE}{9K-E}</math> | ||
| + | | align=center | <math>\tfrac{3(K-\lambda)}{2}</math> | ||
| | | | ||
| + | | align=center | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math> | ||
| | | | ||
| − | | align=center | <math>\tfrac{ | + | | align=center | <math>\tfrac{E}{2(1+\nu)}</math> |
| | | | ||
| − | | align=center | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math> | + | | align=center | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math> |
| | | | ||
| − | |||
| − | |||
| − | |||
| | | | ||
| + | |||
|- | |- | ||
| − | | align=center | <math>\nu=\,</math> | + | |
| − | | align=center | <math>\tfrac{ | + | | align=center | <math>\nu=\,</math> |
| − | + | | align=center | <math>\tfrac{3K-E}{6K}</math> | |
| align=center | <math>\tfrac{\lambda}{3K-\lambda}</math> | | align=center | <math>\tfrac{\lambda}{3K-\lambda}</math> | ||
| − | | align=center | <math>\tfrac{3K-2G}{2(3K+G)}</math> | + | | align=center | <math>\tfrac{3K-2G}{2(3K+G)}</math> |
| | | | ||
| + | | align=center | <math>\tfrac{E}{2G}-1</math> | ||
| | | | ||
| + | | align=center | <math>\tfrac{\lambda}{2(\lambda + G)}</math> | ||
| | | | ||
| | | | ||
| − | |||
| align=center | <math>\tfrac{M - 2G}{2M - 2G}</math> | | align=center | <math>\tfrac{M - 2G}{2M - 2G}</math> | ||
| + | |||
|- | |- | ||
| − | | align=center | <math>M= | + | |
| − | + | | align=center | <math>M=\,</math> | |
| − | | align=center | <math>\tfrac{ | + | | align=center | <math>\tfrac{3K(3K+E)}{9K-E}</math> |
| − | | align=center | <math>3K-2\lambda\,</math> | + | | align=center | <math>3K-2\lambda\,</math> |
| − | | align=center | <math>K+\tfrac{4G}{3}</math> | + | | align=center | <math>K+\tfrac{4G}{3}</math> |
| − | | align=center | <math>\tfrac{ | + | | align=center | <math>\tfrac{3K(1-\nu)}{1+\nu}</math> |
| − | | align=center | <math>\tfrac{ | + | | 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> |
| − | | align=center | <math>\tfrac{ | + | | align=center | <math>\lambda+2G\,</math> |
| − | | align=center | <math>\tfrac{ | + | | align=center | <math>\tfrac{\lambda(1-\nu)}{\nu}</math> |
| + | | align=center | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math> | ||
| | | | ||
|}<noinclude> | |}<noinclude> | ||
Revision as of 18:00, 11 February 2012
| 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>(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>\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>\tfrac{9K(K-\lambda)}{3K-\lambda}</math> | <math>\tfrac{9KG}{3K+G}</math> | <math>3K(1-2\nu)\,</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>K-\tfrac{2G}{3}</math> | <math>\tfrac{3K\nu}{1+\nu}</math> | <math>\tfrac{G(E-2G)}{3G-E}</math> | <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</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>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math> | <math>\tfrac{E}{2(1+\nu)}</math> | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</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>\tfrac{E}{2G}-1</math> | <math>\tfrac{\lambda}{2(\lambda + G)}</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>\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> | |
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
da:Skabelon:Parametre for materialers elasticitet eo:Ŝablono:Elastaj moduloj es:Plantilla:Módulo de elasticidad fa:الگو:مدولهای کشسانی fr:Modules_élastiques </include>