Difference between revisions of "Template:Elastic moduli"

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imported>ShakespeareFan00
(Made table more readable by removing entries of the form K=K which nobody will be looking for.)
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|-
 
|-
 
| style="text-align:center;" | <math>(K,\,E)</math>
 
| style="text-align:center;" | <math>(K,\,E)</math>
| style="text-align:center;" | <math>K</math>
+
| style="text-align:center;" |  
| style="text-align:center;" | <math>E</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{3K(3K-E)}{9K-E}</math>
 
| 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{3KE}{9K-E}</math>
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| style="text-align:center;" | <math>(K,\,\lambda)</math>
 
| style="text-align:center;" | <math>(K,\,\lambda)</math>
| style="text-align:center;" | <math>K</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math>
 
| style="text-align:center;" | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math>
| style="text-align:center;" | <math>\lambda</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{3(K-\lambda)}{2}</math>
 
| 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>\tfrac{\lambda}{3K-\lambda}</math>
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| style="text-align:center;" | <math>(K,\,G)</math>
 
| style="text-align:center;" | <math>(K,\,G)</math>
| style="text-align:center;" | <math>K</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{9KG}{3K+G}</math>
 
| style="text-align:center;" | <math>\tfrac{9KG}{3K+G}</math>
 
| style="text-align:center;" | <math>K-\tfrac{2G}{3}</math>
 
| style="text-align:center;" | <math>K-\tfrac{2G}{3}</math>
| style="text-align:center;" | <math>G</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{3K-2G}{2(3K+G)}</math>
 
| 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+\tfrac{4G}{3}</math>
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| style="text-align:center;" | <math>(K,\,\nu)</math>
 
| style="text-align:center;" | <math>(K,\,\nu)</math>
| style="text-align:center;" | <math>K</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>3K(1-2\nu)\,</math>
 
| 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\nu}{1+\nu}</math>
 
| style="text-align:center;" | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math>
 
| style="text-align:center;" | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math>
| style="text-align:center;" | <math>\nu</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{3K(1-\nu)}{1+\nu}</math>
 
| style="text-align:center;" | <math>\tfrac{3K(1-\nu)}{1+\nu}</math>
 
|
 
|
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| style="text-align:center;" | <math>(K,\,M)</math>
 
| style="text-align:center;" | <math>(K,\,M)</math>
| style="text-align:center;" | <math>K</math>
+
| style="text-align:center;" |
 
| style="text-align:center;" | <math>\tfrac{9K(M-K)}{3K+M}</math>
 
| 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{3K-M}{2}</math>
 
| style="text-align:center;" | <math>\tfrac{3(M-K)}{4}</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;" | <math>\tfrac{3K-M}{3K+M}</math>
| style="text-align:center;" | <math>M</math>
+
| style="text-align:center;" |  
 
|
 
|
  
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| style="text-align:center;" | <math>(E,\,\lambda)</math>
 
| style="text-align:center;" | <math>(E,\,\lambda)</math>
 
| style="text-align:center;" | <math>\tfrac{E + 3\lambda + R}{6}</math>
 
| style="text-align:center;" | <math>\tfrac{E + 3\lambda + R}{6}</math>
| style="text-align:center;" | <math>E</math>
+
| style="text-align:center;" |  
| style="text-align:center;" | <math>\lambda</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{E-3\lambda+R}{4}</math>
 
| 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{2\lambda}{E+\lambda+R}</math>
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| style="text-align:center;" | <math>(E,\,G)</math>
 
| style="text-align:center;" | <math>(E,\,G)</math>
 
| style="text-align:center;" | <math>\tfrac{EG}{3(3G-E)}</math>
 
| style="text-align:center;" | <math>\tfrac{EG}{3(3G-E)}</math>
| style="text-align:center;" | <math>E</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{G(E-2G)}{3G-E}</math>
 
| style="text-align:center;" | <math>\tfrac{G(E-2G)}{3G-E}</math>
| style="text-align:center;" | <math>G</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{E}{2G}-1</math>
 
| style="text-align:center;" | <math>\tfrac{E}{2G}-1</math>
 
| style="text-align:center;" | <math>\tfrac{G(4G-E)}{3G-E}</math>
 
| style="text-align:center;" | <math>\tfrac{G(4G-E)}{3G-E}</math>
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| style="text-align:center;" | <math>(E,\,\nu)</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;" | <math>\tfrac{E}{3(1-2\nu)}</math>
| style="text-align:center;" | <math>E</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\nu}{(1+\nu)(1-2\nu)}</math>
 
| style="text-align:center;" | <math>\tfrac{E}{2(1+\nu)}</math>  
 
| style="text-align:center;" | <math>\tfrac{E}{2(1+\nu)}</math>  
| style="text-align:center;" | <math>\nu</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math>
 
| style="text-align:center;" | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math>
 
|
 
|
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| style="text-align:center;" | <math>(E,\,M)</math>
 
| style="text-align:center;" | <math>(E,\,M)</math>
 
| style="text-align:center;" | <math>\tfrac{3M-E+S}{6}</math>
 
| style="text-align:center;" | <math>\tfrac{3M-E+S}{6}</math>
| style="text-align:center;" | <math>E</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{M-E+S}{4}</math>
 
| 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{3M+E-S}{8}</math>  
 
| style="text-align:center;" | <math>\tfrac{E-M+S}{4M}</math>
 
| style="text-align:center;" | <math>\tfrac{E-M+S}{4M}</math>
| style="text-align:center;" | <math>M</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <p> <math>S=\pm\sqrt{E^2+9M^2-10EM}</math><br />  
 
| style="text-align:center;" | <p> <math>S=\pm\sqrt{E^2+9M^2-10EM}</math><br />  
 
There are two valid solutions.<br />  
 
There are two valid solutions.<br />  
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| style="text-align:center;" | <math>\lambda+ \tfrac{2G}{3}</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;" | <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math>
| style="text-align:center;" | <math>\lambda</math>
+
| style="text-align:center;" |  
| style="text-align:center;" | <math>G</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{\lambda}{2(\lambda + G)}</math>
 
| style="text-align:center;" | <math>\tfrac{\lambda}{2(\lambda + G)}</math>
 
| style="text-align:center;" | <math>\lambda+2G\,</math>
 
| style="text-align:center;" | <math>\lambda+2G\,</math>
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| style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)}{3\nu}</math>
 
| style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)}{3\nu}</math>
 
| style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math>
 
| style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math>
| style="text-align:center;" | <math>\lambda</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math>
 
| style="text-align:center;" | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math>
| style="text-align:center;" | <math>\nu</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{\lambda(1-\nu)}{\nu}</math>
 
| 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>
 
| style="text-align:center;" | Cannot be used when <math>\nu=0 \Leftrightarrow \lambda=0</math>
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| style="text-align:center;" | <math>\tfrac{M + 2\lambda}{3}</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;" | <math>\tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}</math>
| style="text-align:center;" | <math>\lambda</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{M-\lambda}{2}</math>
 
| style="text-align:center;" | <math>\tfrac{M-\lambda}{2}</math>
 
| style="text-align:center;" | <math>\tfrac{\lambda}{M+\lambda}</math>
 
| style="text-align:center;" | <math>\tfrac{\lambda}{M+\lambda}</math>
| style="text-align:center;" | <math>M</math>
+
| style="text-align:center;" |  
 
|
 
|
  
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| style="text-align:center;" | <math>2G(1+\nu)\,</math>
 
| style="text-align:center;" | <math>2G(1+\nu)\,</math>
 
| style="text-align:center;" | <math>\tfrac{2 G \nu}{1-2\nu}</math>
 
| style="text-align:center;" | <math>\tfrac{2 G \nu}{1-2\nu}</math>
| style="text-align:center;" | <math>G</math>
+
| style="text-align:center;" |  
| style="text-align:center;" | <math>\nu</math>
+
| style="text-align:center;" |  
 
| style="text-align:center;" | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math>
 
| style="text-align:center;" | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math>
 
|
 
|
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| style="text-align:center;" | <math>\tfrac{G(3M-4G)}{M-G}</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;" | <math>M - 2G\,</math>
| style="text-align:center;" | <math>G</math>
+
| style="text-align:center;" |
 
| style="text-align:center;" | <math>\tfrac{M - 2G}{2M - 2G}</math>
 
| style="text-align:center;" | <math>\tfrac{M - 2G}{2M - 2G}</math>
| style="text-align:center;" | <math>M</math>
+
| style="text-align:center;" |  
 
|
 
|
  
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| style="text-align:center;" | <math>\tfrac{M \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;" | <math>\tfrac{M(1-2\nu)}{2(1-\nu)}</math>
| style="text-align:center;" | <math>\nu</math>
+
| style="text-align:center;" |  
| style="text-align:center;" | <math>M</math>
+
| style="text-align:center;" |  
 
|
 
|
  

Revision as of 12:25, 29 August 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.
The plus sign leads to <math>\nu\geq 0</math>.

The minus sign leads to <math>\nu\leq 0</math>.

<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