# Lamé parameters

In continuum mechanics, the Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships. 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(not the same units); whereas in the context of elasticity, μ is called the shear modulus,: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, I the identity matrix and tr the trace function. Hooke's law may be written in terms of tensor components using index notation as

$\sigma _{ij}=2\mu E_{ij}+\lambda \delta _{ij}E_{kk},$ where σij is the stress tensor, Eij the strain tensor, and δij the Kronecker delta.

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 = λ + 2/3μ.

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. They have the same dimension as stress and are usually given in the pressure unit [Pa].