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Quantities that connect the components of an elastic stress at some point of a linearly-elastic (or solid deformable) isotropic body with the components of the deformation at this point:
 
Quantities that connect the components of an elastic stress at some point of a linearly-elastic (or solid deformable) isotropic body with the components of the deformation at this point:
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573801.png" /></td> </tr></table>
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\sigma_x = 2 \mu \epsilon_{xx} + \lambda(\epsilon_{xx} + \epsilon_{yy} + \epsilon_{zz}) \ ,
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573802.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573803.png" /> are the normal and tangential constituents of the stress, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573804.png" /> are the components of the deformation and the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573806.png" /> are the Lamé constants. The Lamé constants depend on the material and its temperature. The Lamé constants are connected with the elasticity modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573807.png" /> and Poisson's ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573808.png" /> by
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$$
 
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\tau_{xy} = \mu \epsilon_{xy} \ ,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l0573809.png" /></td> </tr></table>
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$$
 
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where $\sigma$ and $\tau$ are the normal and tangential constituents of the stress, $\epsilon$ are the components of the deformation and the coefficients $\lambda$ and $\mu$ are the Lamé constants. The Lamé constants depend on the material and its temperature. The Lamé constants are connected with the elasticity modulus $E$ and Poisson's ratio $\nu$ by
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l05738010.png" /> is also called Young's modulus and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057380/l05738011.png" /> is the modulus of shear.
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$$
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\mu = G = \frac{E}{2(1+\nu)} \ ,
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$$
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$$
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\lambda = \frac{E_\nu}{(1+\nu)(1-2\nu)} \ ;
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$$
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$E$ is also called Young's modulus and $G$ is the modulus of shear.
  
 
The Lamé constants are named after G. Lamé.
 
The Lamé constants are named after G. Lamé.
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.M. Lifshitz,  "Theory of elasticity" , Pergamon  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.S. [I.S. Sokolnikov] Sokolnikoff,  "Mathematical theory of elasticity" , McGraw-Hill  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.C. Hunter,  "Mechanics of continuous media" , Wiley  (1976)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.M. Lifshitz,  "Theory of elasticity" , Pergamon  (1959)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  I.S. [I.S. Sokolnikov] Sokolnikoff,  "Mathematical theory of elasticity" , McGraw-Hill  (1956)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  S.C. Hunter,  "Mechanics of continuous media" , Wiley  (1976)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Revision as of 16:08, 17 September 2017

Quantities that connect the components of an elastic stress at some point of a linearly-elastic (or solid deformable) isotropic body with the components of the deformation at this point: $$ \sigma_x = 2 \mu \epsilon_{xx} + \lambda(\epsilon_{xx} + \epsilon_{yy} + \epsilon_{zz}) \ , $$ $$ \tau_{xy} = \mu \epsilon_{xy} \ , $$ where $\sigma$ and $\tau$ are the normal and tangential constituents of the stress, $\epsilon$ are the components of the deformation and the coefficients $\lambda$ and $\mu$ are the Lamé constants. The Lamé constants depend on the material and its temperature. The Lamé constants are connected with the elasticity modulus $E$ and Poisson's ratio $\nu$ by $$ \mu = G = \frac{E}{2(1+\nu)} \ , $$ $$ \lambda = \frac{E_\nu}{(1+\nu)(1-2\nu)} \ ; $$ $E$ is also called Young's modulus and $G$ is the modulus of shear.

The Lamé constants are named after G. Lamé.


Comments

References

[a1] E.M. Lifshitz, "Theory of elasticity" , Pergamon (1959) (Translated from Russian)
[a2] I.S. [I.S. Sokolnikov] Sokolnikoff, "Mathematical theory of elasticity" , McGraw-Hill (1956) (Translated from Russian)
[a3] S.C. Hunter, "Mechanics of continuous media" , Wiley (1976)
How to Cite This Entry:
Lamé constants. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_constants&oldid=23362
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article