Physics:Μ(I) rheology

Short description: Model of the rheology of a granular flow

In granular mechanics, the μ(I) rheology is one model of the rheology of a granular flow.

Details

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

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

where [math]\displaystyle{ \dot\gamma }[/math] is the shear rate tensor, [math]\displaystyle{ ||\dot\gamma|| }[/math] 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:

[math]\displaystyle{ \sigma_{ij} = -P\delta_{ij} + \mu(I)P \frac{\dot\gamma_{ij}}{||\dot\gamma||} }[/math]

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 [math]\displaystyle{ \mu(I)P/||\dot\gamma|| }[/math] 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.

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

Development

The μ(I) rheology was developed by Jop et al. in 2006.