Difference between revisions of "Template:Elastic moduli"
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{{Navbox | {{Navbox | ||
| − | | name | + | | 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, \mu</math>) | ||
| + | * [[Poisson's ratio]] (<math>\nu</math>) | ||
| + | * [[P-wave modulus]] (<math>M</math>) | ||
| state = show | | state = show | ||
}} | }} | ||
{| 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=8 | Conversion formulae |
|- | |- | ||
| − | | colspan= | + | | colspan=8 | 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 | + | | style="text-align:center;" | <math>K=\,</math> |
| − | | align | + | | style="text-align:center;" | <math>E=\, </math> |
| − | | align | + | | style="text-align:center;" | <math>\lambda=\,</math> |
| − | + | | style="text-align:center;" | <math>G=\, </math> | |
| − | | | + | | style="text-align:center;" | <math>\nu=\,</math> |
| − | + | | style="text-align:center;" | <math>M=\,</math> | |
| − | | align | + | | style="text-align:center;" | Notes |
| − | + | ||
| − | | align | ||
| − | | align | ||
|- | |- | ||
| − | | align | + | | style="text-align:center;" | <math>(K,\,E)</math> |
| − | | align=center | < | + | | style="text-align:center;" | |
| − | | align | + | | style="text-align:center;" | |
| + | | style="text-align:center;" | <math>\tfrac{3K(3K-E)}{9K-E}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3KE}{9K-E}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3K-E}{6K}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3K(3K+E)}{9K-E}</math> | ||
| | | | ||
| + | |||
| + | |- | ||
| + | |||
| + | | style="text-align:center;" | <math>(K,\,\lambda)</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{3(K-\lambda)}{2}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{\lambda}{3K-\lambda}</math> | ||
| + | | style="text-align:center;" | <math>3K-2\lambda\,</math> | ||
| | | | ||
| − | | align=center | <math>\tfrac{ | + | |
| − | | align=center | <math>\tfrac{2G | + | |- |
| − | | align | + | |
| + | | style="text-align:center;" | <math>(K,\,G)</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{9KG}{3K+G}</math> | ||
| + | | style="text-align:center;" | <math>K-\tfrac{2G}{3}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{3K-2G}{2(3K+G)}</math> | ||
| + | | style="text-align:center;" | <math>K+\tfrac{4G}{3}</math> | ||
| | | | ||
| + | |||
| + | |- | ||
| + | |||
| + | | style="text-align:center;" | <math>(K,\,\nu)</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>3K(1-2\nu)\,</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3K\nu}{1+\nu}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{3K(1-\nu)}{1+\nu}</math> | ||
| | | | ||
| − | + | ||
|- | |- | ||
| − | | align | + | |
| − | | align=center | <math>\tfrac{ | + | | style="text-align:center;" | <math>(K,\,M)</math> |
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{9K(M-K)}{3K+M}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3K-M}{2}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3(M-K)}{4}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3K-M}{3K+M}</math> | ||
| + | | style="text-align:center;" | | ||
| | | | ||
| − | | align=center | <math>\tfrac{ | + | |
| − | | align | + | |- |
| − | | align=center | <math>\ | + | |
| − | | align=center | <math>2G | + | | style="text-align:center;" | <math>(E,\,\lambda)</math> |
| + | | style="text-align:center;" | <math>\tfrac{E + 3\lambda + R}{6}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{E-3\lambda+R}{4}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{2\lambda}{E+\lambda+R}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{E-\lambda+R}{2}</math> | ||
| + | | style="text-align:center;" | <math>R=\sqrt{E^2+9\lambda^2 + 2E\lambda}</math> | ||
| + | |||
| + | |- | ||
| + | |||
| + | | style="text-align:center;" | <math>(E,\,G)</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{EG}{3(3G-E)}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{G(E-2G)}{3G-E}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{E}{2G}-1</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{G(4G-E)}{3G-E}</math> | ||
| | | | ||
| − | | align=center | <math> | + | |
| + | |- | ||
| + | |||
| + | | style="text-align:center;" | <math>(E,\,\nu)</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{E}{3(1-2\nu)}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{E}{2(1+\nu)}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math> | ||
| | | | ||
| − | + | ||
|- | |- | ||
| − | | align=center | <math>\lambda=\,</math> | + | |
| + | | style="text-align:center;" | <math>(E,\,M)</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3M-E+S}{6}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{M-E+S}{4}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{3M+E-S}{8}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{E-M+S}{4M}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>S=\pm\sqrt{E^2+9M^2-10EM}</math><br /> | ||
| + | There are two valid solutions.<br /> | ||
| + | The plus sign leads to <math>\nu\geq 0</math>.<br /> | ||
| + | The minus sign leads to <math>\nu\leq 0</math>.<br /></p> | ||
| + | |||
| + | |- | ||
| + | |||
| + | | style="text-align:center;" | <math>(\lambda,\,G)</math> | ||
| + | | style="text-align:center;" | <math>\lambda+ \tfrac{2G}{3}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{\lambda}{2(\lambda + G)}</math> | ||
| + | | style="text-align:center;" | <math>\lambda+2G\,</math> | ||
| | | | ||
| − | | | + | |
| − | + | |- | |
| − | | align | + | |
| − | | | + | | style="text-align:center;" | <math>(\lambda,\,\nu)</math> |
| − | + | | style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)}{3\nu}</math> | |
| − | | align | + | | style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math> |
| − | | align=center | <math>\tfrac{ | + | | style="text-align:center;" | |
| − | | align=center | <math>\tfrac{ | + | | style="text-align:center;" | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math> |
| − | | align | + | | style="text-align:center;" | |
| + | | style="text-align:center;" | <math>\tfrac{\lambda(1-\nu)}{\nu}</math> | ||
| + | | style="text-align:center;" | Cannot be used when <math>\nu=0 \Leftrightarrow \lambda=0</math> | ||
| + | |||
|- | |- | ||
| − | | align=center | <math> | + | |
| + | | style="text-align:center;" | <math>(\lambda,\,M)</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{M + 2\lambda}{3}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{M-\lambda}{2}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{\lambda}{M+\lambda}</math> | ||
| + | | style="text-align:center;" | | ||
| | | | ||
| − | + | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
|- | |- | ||
| − | | align | + | |
| − | | align | + | | style="text-align:center;" | <math>(G,\,\nu)</math> |
| − | | align | + | | style="text-align:center;" | <math>\tfrac{2G(1+\nu)}{3(1-2\nu)}</math> |
| − | | align | + | | style="text-align:center;" | <math>2G(1+\nu)\,</math> |
| − | | align=center | <math>\tfrac{ | + | | style="text-align:center;" | <math>\tfrac{2 G \nu}{1-2\nu}</math> |
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math> | ||
| | | | ||
| − | + | ||
| − | |||
| − | |||
| − | |||
| − | |||
|- | |- | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | | style="text-align:center;" | <math>(G,\,M)</math> | |
| + | | style="text-align:center;" | <math>M - \tfrac{4G}{3}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{G(3M-4G)}{M-G}</math> | ||
| + | | style="text-align:center;" | <math>M - 2G\,</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | <math>\tfrac{M - 2G}{2M - 2G}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | | ||
| + | |- | ||
| − | == | + | | style="text-align:center;" | <math>(\nu,\,M)</math> |
| − | + | | style="text-align:center;" | <math>\tfrac{M(1+\nu)}{3(1-\nu)}</math> | |
| + | | style="text-align:center;" | <math>\tfrac{M(1+\nu)(1-2\nu)}{1-\nu}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{M \nu}{1-\nu}</math> | ||
| + | | style="text-align:center;" | <math>\tfrac{M(1-2\nu)}{2(1-\nu)}</math> | ||
| + | | style="text-align:center;" | | ||
| + | | style="text-align:center;" | | ||
| + | | | ||
| − | + | |}<noinclude> | |
| − | + | {{Documentation}} | |
| − | + | </noinclude> | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | </ | ||
Latest revision as of 23:16, 1 February 2019
| Conversion formulae | |||||||
|---|---|---|---|---|---|---|---|
| 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> | |||
Usage
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
| The above documentation is transcluded from Template:Elastic moduli/doc. (edit | history) Editors can experiment in this template's sandbox (create | mirror) and testcases (create) pages. Please add categories to the /doc subpage. Subpages of this template. |