Difference between revisions of "Template:Elastic moduli"
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(Made table more readable by removing entries of the form K=K which nobody will be looking for.) |
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* 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}} | ||
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Revision as of 21:25, 3 September 2018
| 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>\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>\tfrac{9K(K-\lambda)}{3K-\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>\tfrac{9KG}{3K+G}</math> | <math>K-\tfrac{2G}{3}</math> | <math>\tfrac{3K-2G}{2(3K+G)}</math> | <math>K+\tfrac{4G}{3}</math> | |||
| <math>(K,\,\nu)</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>\tfrac{3K(1-\nu)}{1+\nu}</math> | |||
| <math>(K,\,M)</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>(E,\,\lambda)</math> | <math>\tfrac{E + 3\lambda + R}{6}</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>\tfrac{G(E-2G)}{3G-E}</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>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math> | <math>\tfrac{E}{2(1+\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>\tfrac{M-E+S}{4}</math> | <math>\tfrac{3M+E-S}{8}</math> | <math>\tfrac{E-M+S}{4M}</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>\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>\tfrac{\lambda(1-2\nu)}{2\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>\tfrac{M-\lambda}{2}</math> | <math>\tfrac{\lambda}{M+\lambda}</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>\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>\tfrac{M - 2G}{2M - 2G}</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> | |||
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