Difference between revisions of "Lamé constants"
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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: | ||
− | + | $$ | |
− | + | \sigma_x = 2 \mu \epsilon_{xx} + \lambda(\epsilon_{xx} + \epsilon_{yy} + \epsilon_{zz}) \ , | |
− | + | $$ | |
− | where | + | $$ |
− | + | \tau_{xy} = \mu \epsilon_{xy} \ , | |
− | + | $$ | |
− | + | 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 the Poisson 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. | The Lamé constants are named after G. Lamé. | ||
Line 17: | Line 23: | ||
====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> | + | <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> | ||
+ | |||
+ | {{TEX|done}} |
Latest revision as of 17:20, 28 December 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 the Poisson 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) |
Lamé constants. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_constants&oldid=15506