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alpha>Indicwiki: Created page with "{{Navbox | name = Elastic moduli | title = Elastic moduli for homogeneous isotropic materials |listclass = hlist | list1 = * Bulk modulus (<math>K<..."
2023-02-24T06:10:24Z
<p>Created page with "{{Navbox | name = Elastic moduli | title = <a href="/index.php?title=Elastic_modulus&action=edit&redlink=1" class="new" title="Elastic modulus (page does not exist)">Elastic moduli</a> for homogeneous <a href="/index.php?title=Isotropic&action=edit&redlink=1" class="new" title="Isotropic (page does not exist)">isotropic</a> materials |listclass = hlist | list1 = * <a href="/index.php?title=Bulk_modulus&action=edit&redlink=1" class="new" title="Bulk modulus (page does not exist)">Bulk modulus</a> (<math>K<..."</p>
<p><b>New page</b></p><div>{{Navbox<br />
| name = Elastic moduli<br />
| title = [[Elastic modulus|Elastic moduli]] for homogeneous [[isotropic]] materials<br />
|listclass = hlist<br />
| list1 =<br />
* [[Bulk modulus]] (<math>K</math>)<br />
* [[Young's modulus]] (<math>E</math>)<br />
* [[Lamé parameters|Lamé's first parameter]] (<math>\lambda</math>)<br />
* [[Shear modulus]] (<math>G, \mu</math>)<br />
* [[Poisson's ratio]] (<math>\nu</math>)<br />
* [[P-wave modulus]] (<math>M</math>)<br />
| state = show<br />
}}<br />
{| class="wikitable mw-collapsible" width="100%" style="font-size:smaller; background:white" align=center<br />
! colspan=8 | Conversion formulae<br />
|-<br />
| 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, provided both for 3D materials (first part of the table) and for 2D materials (second part).<br />
|-<br />
| style="background:#F0F0FF;" | '''3D formulae'''<br />
| style="text-align:center; background:#F0F0FF;" | <math>K=\,</math><br />
| style="text-align:center; background:#F0F0FF;" | <math>E=\, </math><br />
| style="text-align:center; background:#F0F0FF;" | <math>\lambda=\,</math><br />
| style="text-align:center; background:#F0F0FF;" | <math>G=\, </math><br />
| style="text-align:center; background:#F0F0FF;" | <math>\nu=\,</math><br />
| style="text-align:center; background:#F0F0FF;" | <math>M=\,</math><br />
| style="text-align:center; background:#F0F0FF;" | Notes<br />
<br />
|-<br />
| style="text-align:center;" | <math>(K,\,E)</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{3K(3K-E)}{9K-E}</math><br />
| style="text-align:center;" | <math>\tfrac{3KE}{9K-E}</math><br />
| style="text-align:center;" | <math>\tfrac{3K-E}{6K}</math><br />
| style="text-align:center;" | <math>\tfrac{3K(3K+E)}{9K-E}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(K,\,\lambda)</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{9K(K-\lambda)}{3K-\lambda}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{3(K-\lambda)}{2}</math><br />
| style="text-align:center;" | <math>\tfrac{\lambda}{3K-\lambda}</math><br />
| style="text-align:center;" | <math>3K-2\lambda\,</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(K,\,G)</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{9KG}{3K+G}</math><br />
| style="text-align:center;" | <math>K-\tfrac{2G}{3}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{3K-2G}{2(3K+G)}</math><br />
| style="text-align:center;" | <math>K+\tfrac{4G}{3}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(K,\,\nu)</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>3K(1-2\nu)\,</math><br />
| style="text-align:center;" | <math>\tfrac{3K\nu}{1+\nu}</math><br />
| style="text-align:center;" | <math>\tfrac{3K(1-2\nu)}{2(1+\nu)}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{3K(1-\nu)}{1+\nu}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(K,\,M)</math><br />
| style="text-align:center;" |<br />
| style="text-align:center;" | <math>\tfrac{9K(M-K)}{3K+M}</math><br />
| style="text-align:center;" | <math>\tfrac{3K-M}{2}</math><br />
| style="text-align:center;" | <math>\tfrac{3(M-K)}{4}</math><br />
| style="text-align:center;" | <math>\tfrac{3K-M}{3K+M}</math><br />
| style="text-align:center;" | <br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(E,\,\lambda)</math><br />
| style="text-align:center;" | <math>\tfrac{E + 3\lambda + R}{6}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{E-3\lambda+R}{4}</math><br />
| style="text-align:center;" | <math>\tfrac{2\lambda}{E+\lambda+R}</math><br />
| style="text-align:center;" | <math>\tfrac{E-\lambda+R}{2}</math><br />
| style="text-align:center;" | <math>R=\sqrt{E^2+9\lambda^2 + 2E\lambda}</math><br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(E,\,G)</math><br />
| style="text-align:center;" | <math>\tfrac{EG}{3(3G-E)}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{G(E-2G)}{3G-E}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{E}{2G}-1</math><br />
| style="text-align:center;" | <math>\tfrac{G(4G-E)}{3G-E}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(E,\,\nu)</math><br />
| style="text-align:center;" | <math>\tfrac{E}{3(1-2\nu)}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{E\nu}{(1+\nu)(1-2\nu)}</math><br />
| style="text-align:center;" | <math>\tfrac{E}{2(1+\nu)}</math> <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(E,\,M)</math><br />
| style="text-align:center;" | <math>\tfrac{3M-E+S}{6}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{M-E+S}{4}</math><br />
| style="text-align:center;" | <math>\tfrac{3M+E-S}{8}</math> <br />
| style="text-align:center;" | <math>\tfrac{E-M+S}{4M}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>S=\pm\sqrt{E^2+9M^2-10EM}</math><br /> <br />
There are two valid solutions.<br /> <br />
The plus sign leads to <math>\nu\geq 0</math>.<br /> <br />
The minus sign leads to <math>\nu\leq 0</math>.<br /></p><br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(\lambda,\,G)</math><br />
| style="text-align:center;" | <math>\lambda+ \tfrac{2G}{3}</math><br />
| style="text-align:center;" | <math>\tfrac{G(3\lambda + 2G)}{\lambda + G}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{\lambda}{2(\lambda + G)}</math><br />
| style="text-align:center;" | <math>\lambda+2G\,</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(\lambda,\,\nu)</math><br />
| style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)}{3\nu}</math><br />
| style="text-align:center;" | <math>\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{\lambda(1-2\nu)}{2\nu}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{\lambda(1-\nu)}{\nu}</math><br />
| style="text-align:center;" | Cannot be used when <math>\nu=0 \Leftrightarrow \lambda=0</math><br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(\lambda,\,M)</math><br />
| style="text-align:center;" | <math>\tfrac{M + 2\lambda}{3}</math><br />
| style="text-align:center;" | <math>\tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{M-\lambda}{2}</math><br />
| style="text-align:center;" | <math>\tfrac{\lambda}{M+\lambda}</math><br />
| style="text-align:center;" | <br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(G,\,\nu)</math><br />
| style="text-align:center;" | <math>\tfrac{2G(1+\nu)}{3(1-2\nu)}</math><br />
| style="text-align:center;" | <math>2G(1+\nu)\,</math><br />
| style="text-align:center;" | <math>\tfrac{2 G \nu}{1-2\nu}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{2G(1-\nu)}{1-2\nu} </math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(G,\,M)</math><br />
| style="text-align:center;" | <math>M - \tfrac{4G}{3}</math><br />
| style="text-align:center;" | <math>\tfrac{G(3M-4G)}{M-G}</math><br />
| style="text-align:center;" | <math>M - 2G\,</math><br />
| style="text-align:center;" |<br />
| style="text-align:center;" | <math>\tfrac{M - 2G}{2M - 2G}</math><br />
| style="text-align:center;" | <br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(\nu,\,M)</math><br />
| style="text-align:center;" | <math>\tfrac{M(1+\nu)}{3(1-\nu)}</math><br />
| style="text-align:center;" | <math>\tfrac{M(1+\nu)(1-2\nu)}{1-\nu}</math><br />
| style="text-align:center;" | <math>\tfrac{M \nu}{1-\nu}</math><br />
| style="text-align:center;" | <math>\tfrac{M(1-2\nu)}{2(1-\nu)}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <br />
|<br />
<br />
|-<br />
| style="background:#F0F0FF;" | '''2D formulae'''<br />
| style="text-align:center; background:#F0F0FF;" | <math>K_\mathrm{2D}=\,</math><br />
| style="text-align:center; background:#F0F0FF;" | <math>E_\mathrm{2D}=\, </math><br />
| style="text-align:center; background:#F0F0FF;" | <math>\lambda_\mathrm{2D}=\,</math><br />
| style="text-align:center; background:#F0F0FF;" | <math>G_\mathrm{2D}=\, </math><br />
| style="text-align:center; background:#F0F0FF;" | <math>\nu_\mathrm{2D}=\,</math><br />
| style="text-align:center; background:#F0F0FF;" | <math>M_\mathrm{2D}=\,</math><br />
| style="text-align:center; background:#F0F0FF;" | Notes<br />
<br />
|-<br />
| style="text-align:center;" | <math>(K_\mathrm{2D},\,E_\mathrm{2D})</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{2K_\mathrm{2D}(2K_\mathrm{2D}-E_\mathrm{2D})}{4K_\mathrm{2D}-E_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>\tfrac{K_\mathrm{2D}E_\mathrm{2D}}{4K_\mathrm{2D}-E_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>\tfrac{2K_\mathrm{2D}-E_\mathrm{2D}}{2K_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>\tfrac{4K_\mathrm{2D}^2}{4K_\mathrm{2D}-E_\mathrm{2D}}</math><br />
|<br />
<br />
|-<br />
| style="text-align:center;" | <math>(K_\mathrm{2D},\,\lambda_\mathrm{2D})</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{4K_\mathrm{2D}(K_\mathrm{2D}-\lambda_\mathrm{2D})}{2K_\mathrm{2D}-\lambda_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>K_\mathrm{2D}-\lambda_\mathrm{2D}</math><br />
| style="text-align:center;" | <math>\tfrac{\lambda_\mathrm{2D}}{2K_\mathrm{2D}-\lambda_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>2K_\mathrm{2D}-\lambda_\mathrm{2D}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(K_\mathrm{2D},\,G_\mathrm{2D})</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{4K_\mathrm{2D}G_\mathrm{2D}}{K_\mathrm{2D}+G_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>K_\mathrm{2D}-G_\mathrm{2D}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{K_\mathrm{2D}-G_\mathrm{2D}}{K_\mathrm{2D}+G_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>K_\mathrm{2D}+G_\mathrm{2D}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(K_\mathrm{2D},\,\nu_\mathrm{2D})</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>2K_\mathrm{2D}(1-\nu_\mathrm{2D})\,</math><br />
| style="text-align:center;" | <math>\tfrac{2K_\mathrm{2D}\nu_\mathrm{2D}}{1+\nu_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>\tfrac{K_\mathrm{2D}(1-\nu_\mathrm{2D})}{1+\nu_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{2K_\mathrm{2D}}{1+\nu_\mathrm{2D}}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(E_\mathrm{2D},\,G_\mathrm{2D})</math><br />
| style="text-align:center;" | <math>\tfrac{E_\mathrm{2D}G_\mathrm{2D}}{4G_\mathrm{2D}-E_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{2G_\mathrm{2D}(E_\mathrm{2D}-2G_\mathrm{2D})}{4G_\mathrm{2D}-E_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{E_\mathrm{2D}}{2G_\mathrm{2D}}-1</math><br />
| style="text-align:center;" | <math>\tfrac{4G_\mathrm{2D}^2}{4G_\mathrm{2D}-E_\mathrm{2D}}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(E_\mathrm{2D},\,\nu_\mathrm{2D})</math><br />
| style="text-align:center;" | <math>\tfrac{E_\mathrm{2D}}{2(1-\nu_\mathrm{2D})}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{E_\mathrm{2D}\nu_\mathrm{2D}}{(1+\nu_\mathrm{2D})(1-\nu_\mathrm{2D})}</math><br />
| style="text-align:center;" | <math>\tfrac{E_\mathrm{2D}}{2(1+\nu_\mathrm{2D})}</math> <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{E_\mathrm{2D}}{(1+\nu_\mathrm{2D})(1-\nu_\mathrm{2D})}</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(\lambda_\mathrm{2D},\,G_\mathrm{2D})</math><br />
| style="text-align:center;" | <math>\lambda_\mathrm{2D}+ G_\mathrm{2D}</math><br />
| style="text-align:center;" | <math>\tfrac{4G_\mathrm{2D}(\lambda_\mathrm{2D} + G_\mathrm{2D})}{\lambda_\mathrm{2D} + 2G_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{\lambda_\mathrm{2D}}{\lambda_\mathrm{2D} + 2G_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>\lambda_\mathrm{2D}+2G_\mathrm{2D}\,</math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(\lambda_\mathrm{2D},\,\nu_\mathrm{2D})</math><br />
| style="text-align:center;" | <math>\tfrac{\lambda_\mathrm{2D}(1+\nu_\mathrm{2D})}{2\nu_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>\tfrac{\lambda_\mathrm{2D}(1+\nu_\mathrm{2D})(1-\nu_\mathrm{2D})}{\nu_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{\lambda_\mathrm{2D}(1-\nu_\mathrm{2D})}{2\nu_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{\lambda_\mathrm{2D}}{\nu_\mathrm{2D}}</math><br />
| style="text-align:center;" | Cannot be used when <math>\nu_\mathrm{2D}=0 \Leftrightarrow \lambda_\mathrm{2D}=0</math><br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(G_\mathrm{2D},\,\nu_\mathrm{2D})</math><br />
| style="text-align:center;" | <math>\tfrac{G_\mathrm{2D}(1+\nu_\mathrm{2D})}{1-\nu_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>2G_\mathrm{2D}(1+\nu_\mathrm{2D})\,</math><br />
| style="text-align:center;" | <math>\tfrac{2 G_\mathrm{2D} \nu_\mathrm{2D}}{1-\nu_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <br />
| style="text-align:center;" | <math>\tfrac{2G_\mathrm{2D}}{1-\nu_\mathrm{2D}} </math><br />
|<br />
<br />
|-<br />
<br />
| style="text-align:center;" | <math>(G_\mathrm{2D},\,M_\mathrm{2D})</math><br />
| style="text-align:center;" | <math>M_\mathrm{2D} - G_\mathrm{2D}</math><br />
| style="text-align:center;" | <math>\tfrac{4G_\mathrm{2D}(M_\mathrm{2D}-G_\mathrm{2D})}{M_\mathrm{2D}}</math><br />
| style="text-align:center;" | <math>M_\mathrm{2D} - 2G_\mathrm{2D}\,</math><br />
| style="text-align:center;" |<br />
| style="text-align:center;" | <math>\tfrac{M_\mathrm{2D} - 2G_\mathrm{2D}}{M_\mathrm{2D}}</math><br />
| style="text-align:center;" | <br />
|<br />
<br />
<br />
<br />
<br />
<br />
|}<noinclude><br />
{{Documentation}}<br />
</noinclude></div>
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