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,

$\boldsymbol{\sigma}=2\mu \boldsymbol{\varepsilon} +\lambda \; \mathrm{tr}(\boldsymbol{\varepsilon})I$

where σ is the stress, ε the strain tensor, $\scriptstyle I$ the identity matrix and $\scriptstyle\mathrm{tr}(\cdot)$ 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 $K = \lambda + (2/3)\mu$.

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

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

Conversion formulas
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.
	$K=\,$	$E=\,$	$\lambda=\,$	$G=\,$	$\nu=\,$	$M=\,$	Notes
$(K,\,E)$	$K$	$E$	$\tfrac{3K(3K-E)}{9K-E}$	$\tfrac{3KE}{9K-E}$	$\tfrac{3K-E}{6K}$	$\tfrac{3K(3K+E)}{9K-E}$
$(K,\,\lambda)$	$K$	$\tfrac{9K(K-\lambda)}{3K-\lambda}$	$\lambda$	$\tfrac{3(K-\lambda)}{2}$	$\tfrac{\lambda}{3K-\lambda}$	$3K-2\lambda\,$
$(K,\,G)$	$K$	$\tfrac{9KG}{3K+G}$	$K-\tfrac{2G}{3}$	$G$	$\tfrac{3K-2G}{2(3K+G)}$	$K+\tfrac{4G}{3}$
$(K,\,\nu)$	$K$	$3K(1-2\nu)\,$	$\tfrac{3K\nu}{1+\nu}$	$\tfrac{3K(1-2\nu)}{2(1+\nu)}$	$\nu$	$\tfrac{3K(1-\nu)}{1+\nu}$
$(K,\,M)$	$K$	$\tfrac{9K(M-K)}{3K+M}$	$\tfrac{3K-M}{2}$	$\tfrac{3(M-K)}{4}$	$\tfrac{3K-M}{3K+M}$	$M$
$(E,\,\lambda)$	$\tfrac{E + 3\lambda + R}{6}$	$E$	$\lambda$	$\tfrac{E-3\lambda+R}{4}$	$\tfrac{2\lambda}{E+\lambda+R}$	$\tfrac{E-\lambda+R}{2}$	$R=\sqrt{E^2+9\lambda^2 + 2E\lambda}$
$(E,\,G)$	$\tfrac{EG}{3(3G-E)}$	$E$	$\tfrac{G(E-2G)}{3G-E}$	$G$	$\tfrac{E}{2G}-1$	$\tfrac{G(4G-E)}{3G-E}$
$(E,\,\nu)$	$\tfrac{E}{3(1-2\nu)}$	$E$	$\tfrac{E\nu}{(1+\nu)(1-2\nu)}$	$\tfrac{E}{2(1+\nu)}$	$\nu$	$\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}$
$(E,\,M)$	$\tfrac{3M-E+S}{6}$	$E$	$\tfrac{M-E+S}{4}$	$\tfrac{3M+E-S}{8}$	$\tfrac{E-M+S}{4M}$	$M$	$S=\pm\sqrt{E^2+9M^2-10EM}$ There are two valid solutions. The plus sign leads to $\nu\geq 0$. The minus sign leads to $\nu\leq 0$.
$(\lambda,\,G)$	$\lambda+ \tfrac{2G}{3}$	$\tfrac{G(3\lambda + 2G)}{\lambda + G}$	$\lambda$	$G$	$\tfrac{\lambda}{2(\lambda + G)}$	$\lambda+2G\,$
$(\lambda,\,\nu)$	$\tfrac{\lambda(1+\nu)}{3\nu}$	$\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$\lambda$	$\tfrac{\lambda(1-2\nu)}{2\nu}$	$\nu$	$\tfrac{\lambda(1-\nu)}{\nu}$	Cannot be used when $\nu=0 \Leftrightarrow \lambda=0$
$(\lambda,\,M)$	$\tfrac{M + 2\lambda}{3}$	$\tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}$	$\lambda$	$\tfrac{M-\lambda}{2}$	$\tfrac{\lambda}{M+\lambda}$	$M$
$(G,\,\nu)$	$\tfrac{2G(1+\nu)}{3(1-2\nu)}$	$2G(1+\nu)\,$	$\tfrac{2 G \nu}{1-2\nu}$	$G$	$\nu$	$\tfrac{2G(1-\nu)}{1-2\nu}$
$(G,\,M)$	$M - \tfrac{4G}{3}$	$\tfrac{G(3M-4G)}{M-G}$	$M - 2G\,$	$G$	$\tfrac{M - 2G}{2M - 2G}$	$M$
$(\nu,\,M)$	$\tfrac{M(1+\nu)}{3(1-\nu)}$	$\tfrac{M(1+\nu)(1-2\nu)}{1-\nu}$	$\tfrac{M \nu}{1-\nu}$	$\tfrac{M(1-2\nu)}{2(1-\nu)}$	$\nu$	$M$