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 typically denoted by G instead of μ.

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

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