Constitutive equation

Remarks

For an isotropic linearly elastic solid with Lamé constants $\lambda$ and $\mu$, the constitutive equation $\sigma = \lambda (\operatorname{tr} \varepsilon) I + 2 \mu \varepsilon$ links stress to the infinitesimal strain $\varepsilon$. For a Newtonian fluid, $\sigma = -p I + \lambda (\operatorname{div} v) I + 2 \mu D$, where $D$ is the symmetric part of the velocity gradient in the Eulerian description and $p$ the pressure. The balance of linear momentum $\rho\, Dv/Dt = \operatorname{div} \sigma + \rho b$ is not a constitutive equation: it holds for every material. Admissible constitutive equations are restricted by general principles, chiefly material frame-indifference and the second law of thermodynamics.