Rheology

Remarks

Example: the Newtonian fluid law $\sigma = -p I + 2 \mu D$, where $D$ is the strain rate tensor and $\mu$ is a constant, is a constitutive equation. It uses only the present value of the deformation rate. The upper-convected Maxwell model is a constitutive equation in which the stress depends on the whole past of the deformation.

Non-example: the balance of momentum $\rho \, Du/Dt = \nabla \cdot \sigma + f$ is not a constitutive equation. It holds for every continuous material and says nothing about how $\sigma$ depends on the deformation.

The word is mostly used for materials that are neither ideal elastic solids nor Newtonian fluids.