μ(I) rheology

The inertial number of a granular flow is a dimensionless quantity defined as

${\displaystyle I={\frac {||{\dot {\gamma }}||d}{\sqrt {P/\rho }}},}$

where ${\displaystyle {\dot {\gamma }}}$ is the shear rate tensor, ${\displaystyle ||{\dot {\gamma }}||}$ is its magnitude, d is the average particle diameter, P is the isotropic pressure and ρ is the density. It is a local quantity and may take different values at different locations in the flow.

The μ(I) rheology asserts a constitutive relationship between the stress tensor of the flow and the rate of strain tensor:

${\displaystyle \sigma _{ij}=-P\delta _{ij}+\mu (I)P{\frac {{\dot {\gamma }}_{ij}}{||{\dot {\gamma }}||}}}$

where the eponymous μ(I) is a dimensionless function of I. As with Newtonian fluids, the first term -Pδ_ij represents the effect of pressure. The second term represents a shear stress: it acts in the direction of the shear, and its magnitude is equal to the pressure multiplied by a coefficient of friction μ(I). This is therefore a generalisation of the standard Coulomb friction model. The multiplicative term ${\displaystyle \mu (I)P/||{\dot {\gamma }}||}$ can be interpreted as the effective viscosity of the granular material, which tends to infinity in the limit of vanishing shear flow, ensuring the existence of a yield criterion.[1]

One deficiency of the μ(I) rheology is that it does not capture the hysteretic properties of a granular material.[2]

The μ(I) rheology was developed by Jop et al. in 2006.[1][3]