Material law describing a non viscous fluid.

Stresses are computed with

$$ \sigma_{ij} = s_{ij} + \delta_{ij} p $$ with $ s_{ij} = 0 $ in a non viscous fluid.

The equation which associates pressure and volume is $$ dP = -K \frac{dV}{V} $$ where $K$ is the bulk modulus.

Norton-Hoff law descriding a viscous fluid.

Stresses are computed with $$ \sigma_{ij} = S_{ij} + \delta_{ij} P $$ The following equation which associates the stress deviator tensor ($ s_{ij} $) and the strain rate deviator tensor ($ D_{ij} $) is

$$ S_{ij} = 2 \mu D_{ij} \left( \sqrt{3} \ \sqrt{\frac{2}{3} D_{lm}.D_{lm}} \right)^{m-1}$$

For a newtonian fluid : $ m=1 \rightarrow S_{ij} = 2 \mu D_{ij} $

The equation which associates pressure and volume is $$ dP = K \frac{dV}{V} $$ where $K$ is the bulk modulus.

Norton-Hoff law including thermal aspects.