# Lamé parameters

In continuum mechanics, the **Lamé parameters** (also called the **Lamé coefficients** or **Lamé constants**) are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships.^{[1]} In general, λ and μ are individually referred to as *Lamé's first parameter* and *Lamé's second parameter*, respectively. Other names are sometimes employed for one or both parameters, depending on context. For example, the parameter μ is referred to in fluid dynamics as the dynamic viscosity of a fluid; whereas in the context of elasticity, μ is called the shear modulus,^{[2]}^{:p.333} and is sometimes denoted by *G* instead of μ. Typically the notation G is seen paired with the use of Young's modulus, and the notation μ is paired with the use of λ.

In homogeneous and isotropic materials, these define Hooke's law in 3D,

where σ is the stress, ε the strain tensor, the identity matrix and the trace function.

The two parameters together constitute a parameterization of the elastic moduli for homogeneous isotropic media, popular in mathematical literature, and are thus related to the other elastic moduli; for instance, the bulk modulus can be expressed as .

Although the shear modulus, μ, must be positive, the Lamé's first parameter, λ, can be negative, in principle; however, for most materials it is also positive.

The parameters are named after Gabriel Lamé.

## Further reading

- K. Feng, Z.-C. Shi,
*Mathematical Theory of Elastic Structures*, Springer New York, ISBN 0-387-51326-4, (1981) - G. Mavko, T. Mukerji, J. Dvorkin,
*The Rock Physics Handbook*, Cambridge University Press (paperback), ISBN 0-521-54344-4, (2003) - W.S. Slaughter,
*The Linearized Theory of Elasticity*, Birkhäuser, ISBN 0-8176-4117-3, (2002)

## References

- ↑ "Lamé Constants". Weisstein, Eric. Eric Weisstein's World of Science, A Wolfram Web Resource. Retrieved 2015-02-22.
- ↑ Jean Salencon (2001), "Handbook of Continuum Mechanics: General Concepts, Thermoelasticity". Springer Science & Business Media ISBN 3-540-41443-6

Conversion formulas | |||||||
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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. | |||||||

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Cannot be used when | |||||||