title: Μ(I) rheology
text: # Μ(I) rheology
  
  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
  
  $I={\frac {||{\dot {\gamma }}||d}{\sqrt {P/\rho }}},$
  
  where ${\dot {\gamma }}$ is the shear rate tensor, $||{\dot {\gamma }}||$ 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:
  
  $\sigma _{ij}=-P\delta _{ij}+\mu (I)P{\frac {{\dot {\gamma }}_{ij}}{||{\dot {\gamma }}||}}$
  
  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 $\mu (I)P/||{\dot {\gamma }}||$ 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.
  
  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.
  
  