Difference between revisions of "Template:Elastic moduli"
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imported>YiFeiBot m (Bot: Migrating interwiki links, now provided by Wikidata on d:q11024927) |
imported>Xenios~dewiki m (Added mu to Navbox. Since mu is often used for the shear modulus instead of G, there should be some reference to it.) |
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* [[Young's modulus]] (<math>E</math>) | * [[Young's modulus]] (<math>E</math>) | ||
* [[Lamé parameters|Lamé's first parameter]] (<math>\lambda</math>) | * [[Lamé parameters|Lamé's first parameter]] (<math>\lambda</math>) | ||
| − | * [[Shear modulus]] (<math>G</math>) | + | * [[Shear modulus]] (<math>G, \mu</math>) |
* [[Poisson's ratio]] (<math>\nu</math>) | * [[Poisson's ratio]] (<math>\nu</math>) | ||
* [[P-wave modulus]] (<math>M</math>) | * [[P-wave modulus]] (<math>M</math>) | ||
Revision as of 21:01, 19 July 2015
| 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,\,\lambda)</math> | <math>\tfrac{E + 3\lambda + R}{6}</math> | <math>E</math> | <math>\lambda</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>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>(E,\,M)</math> | <math>\tfrac{3M-E+S}{6}</math> | <math>E</math> | <math>\tfrac{M-E+S}{4}</math> | <math>\tfrac{3M+E-S}{8}</math> | <math>\tfrac{E-M+S}{4M}</math> | <math>M</math> | <math>S=\pm\sqrt{E^2+9M^2-10EM}</math> There are two valid solutions. |
| <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 \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>\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