Lamé parameters

In homogeneous and isotropic materials, these define Hooke's law in 3D, $${\displaystyle \mathit{\sigma }=2\mu \mathit{\epsilon }+\lambda \mathrm{tr}(\mathit{\epsilon })I,}$$ where σ is the stress tensor, 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 $${\displaystyle {\sigma }_{ij}=2\mu {\epsilon }_{ij}+\lambda {\delta }_{ij}{\epsilon }_{kk},}$$ where δ_ij is 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. Relations for other moduli are found in the (λ, G) row of the conversions table at the end of this article.

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 SI unit of stress [Pa].