# Lamé constants

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) |

**How to Cite This Entry:**

Lamé constants.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Lam%C3%A9_constants&oldid=42630