In granular mechanics, the ''μ''(''I'') rheology is one model of the

rheology Rheology (; ) is the study of the flow of matter, primarily in a fluid (liquid or gas) state but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applie ...

of a granular flow.

Details

The inertial number of a granular flow is a

dimensionless quantity Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...

defined as

I = \frac,

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_ = -P\delta_ + \mu(I)P \frac

where the eponymous ''μ''(''I'') is a dimensionless function of ''I''. As with

Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of cha ...

s, the first term -''Pδ''_''ij'' represents the effect of pressure. The second term represents a

shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...

: 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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...

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 Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...

properties of a granular material.

Development

The ''μ''(''I'') rheology was developed by Pierre Jop ''et al.'' in 2006. Since its initial introduction, many works has been carried out to modify and improve this rheology model. This model provides an alternative approach to the Discrete Element Method (DEM), offering a lower computational cost for simulating granular flows within mixers.

References

{{DEFAULTSORT:Mu I rheology Rheology Granularity of materials

Details

Development

See also

References