Difference between revisions of "Template:Elastic moduli"
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| 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>) | + | |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</math>) | ||
| + | * [[Poisson's ratio]] (<math>\nu</math>) | ||
| + | * [[P-wave modulus]] (<math>M</math>) | ||
| state = show | | state = show | ||
}} | }} | ||
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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. | 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== | ==References== | ||
* G. Mavko, T. Mukerji, J. Dvorkin. ''The Rock Physics Handbook''. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4 | * G. Mavko, T. Mukerji, J. Dvorkin. ''The Rock Physics Handbook''. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4 | ||
| − | [[Category:Physics templates | + | [[Category:Physics templates]] |
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[[da:Skabelon:Parametre for materialers elasticitet]] | [[da:Skabelon:Parametre for materialers elasticitet]] | ||
[[es:Plantilla:Módulo de elasticidad]] | [[es:Plantilla:Módulo de elasticidad]] | ||
Revision as of 21:07, 18 September 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>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>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{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>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{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>\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
da:Skabelon:Parametre for materialers elasticitet es:Plantilla:Módulo de elasticidad eo:Ŝablono:Elastaj moduloj fa:الگو:مدولهای کشسانی fr:Modèle:Modules élastiques zh:Template:弹性模量