Well-posed continuum equations for granular flow with compressibility and μ(I)-rheology

What is it about?

Continuum modelling of granular flow has been plagued with the issue of ill-posed dynamic equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent μ(I)-rheology is ill-posed when the non-dimensional inertial number I is too high or too low. Here, incorporating ideas from critical-state soil mechanics, we derive conditions for well-posedness of partial differential equations that combine compressibility with I-dependent rheology. When the I-dependence comes from a specific friction coefficient μ(I), our results show that, with compressibility, the equations are well-posed for all deformation rates provided that μ(I) satisfies certain minimal, physically natural, inequalities.

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Why is it important?

This is the first paper to combine the μ(I)-rheology with the concepts of critical-state soil mechanics. By purposefully leaving the functions, f and Y, responsible for shear-driven and compression-driven dissipation, as general as possible, the paper introduces a very broad framework. It is shown that a certain simple choice for these functions allows for both well-posedness and for a match with the μ(I)-rheology in the limit that the volume fraction in a flow is no longer changing (iso-choric deformations). Steady solutions of the resultant compressible equations are reassuringly close to those for flow down inclined frictional planes with the original incompressible model. This means that many existing experimental results may be matched in addition to the prediction of bulk density variations in a flow. Details of the transition between the solid-like and liquid-like behaviour of granular materials is a long-standing area of investigation. What is known is that the volume fraction plays a vital role in both regimes. Having dynamic compressible equations, with broad generality and the possibility of well-posedness, is a vital step on the road to understanding these complex and disparate physical phenomena.

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